Gaussian fluctuations for the two-urn model

Abstract

We introduce a modification of the generalized Pólya urn model containing two urns, and we study the number of balls $B_j(n)$ of a given color $j\in\{1,\ldots,J\}$ added to the urns after n draws, where $J\in\mathbb{N}$. We provide sufficient conditions under which the random variables $(B_j(n))_{n\in\mathbb{N}}$, properly normalized and centered, converge weakly to a limiting random variable. The result reveals a similar trichotomy as in the classical case with one urn, one of the main differences being that in the scaling we encounter 1-periodic continuous functions. Another difference in our results compared to the classical urn models is that the phase transition of the second-order behavior occurs at $\sqrt{\rho}$ and not at $\rho/2$, where $\rho$ is the dominant eigenvalue of the mean replacement matrix.

1. Introduction

A brief overview of classical Pólya urn models. For $J\in\mathbb{N}$ , a J-dimensional Pólya urn process $(B(n))_{n\in\mathbb{N}}$ is an $\mathbb{N}^J$ -valued stochastic process which represents the evolution of an urn containing balls of J different colors denoted by $1,2,\ldots, J$ . The initial composition of the urn can be specified by a J-dimensional vector B(0) given by $B(0)=\big(B_1(0),\ldots,B_J(0)\big)$ , the jth coordinate $B_j(0)$ of B(0) representing the number of balls of color j present in the urn at the beginning of the process, i.e. at time 0. At each subsequent time step $n\geq 1$ , we pick a ball uniformly at random, inspect its color, and put it back into the urn together with a random collection of additional balls, whose colors are given by $L^{(j)}=(L^{(j,1)},\ldots,L^{(j,J)})$ if the selected ball has color j (which happens with probability proportional to the number of balls of color j already present in the urn). This rule for adding balls can be summed up by the so-called replacement matrix L, which in our case is a random matrix L defined as follows. For $J\in\mathbb{N}$ , we write $[J]\;:\!=\;\{1,\ldots,J\}$ , and we consider a sequence $(L^{(j)})_{j\in [J]}$ of J independent $\mathbb{N}^J$ -valued random (column) vectors. We denote by L the $J\times J$ random matrix with column vectors $L^{(j)}$ , so $L=\left(L^{(1)},L^{(2)},\ldots,L^{(J)}\right)$ , and by $a_{ij}=\mathbb{E}\left[L^{(j,i)}\right]$ the expectation of $L^{(j,i)}$ , for all $i,j\in [J]$ ; finally, we let $A=(a_{ij})_{i,j\in[J]}$ , so that $\mathbb{E} L=A$ . We then continue the process, each time taking an independent (of everything else) copy of the replacement matrix L. Note that the model described involves replacement, meaning that the selected ball is placed back into the urn after each draw. However, it is also possible to study a model without replacement by considering the replacement matrix $L-I$ , where I is the $J\times J$ identity matrix. In this case, it might happen that some diagonal entries of $L-I$ are equal to $-1$ , which means a ball is removed from the urn. The urn process is the sequence $(B(n))_{n\geq 1}$ of J-dimensional random vectors with nonnegative integer coordinates, and the jth coordinate $B_j(n)$ of B(n) represents the number of balls of color j in the urn after the nth draw, for $j\in [J]$ . We also define $B^{\circ}(n)$ to be the number of balls drawn up to time n; that is, $B^\circ_j(n)$ represents the number of balls of color j drawn up to the nth draw (in particular, $B^\circ_1(n)+\cdots+B^\circ_J(n)=n$ ). As one expects, the limit behavior of $(B(n))_{n\geq 1}$ and $(B^\circ(n))_{n\ge 1}$ depends on the distribution of the replacement matrix L, and in particular on the spectral properties of its mean value matrix A.

The literature on limit theorems for Pólya urn models is enormous and any attempt to give a complete survey here is hopeless, but we mention some relevant references and results in this direction. For additional results, the reader is referred to the cited articles and the references therein. In 1930, in his original article [10], Pólya investigates a two-color urn process with replacement matrix L being the identity. If L is a non-random, irreducible matrix with exclusively nonnegative entries, then it is well established that the sequence $B^\circ(n)/n$ converges almost surely to u as n goes to infinity, where u is the left eigenvector associated with $\rho$ , the spectral radius of $A=\mathbb{E} L$ . The coordinates of u are all nonnegative and normalized in such a way that they sum up to one; see [2, 5, 9, 12] for more details. The second-order behavior of the sequence $(B^\circ(n))_{n\in\mathbb{N}}$ depends on the second eigenvalue $\lambda_2$ (ordered by real parts) of A. If $\textrm{Re}(\lambda_2)$ , the real part of the second-largest eigenvalue, is less than or equal to $\rho/2$ , then the fluctuations around the limit u are Gaussian (with a random variance). The magnitude of the fluctuations is of order $\sqrt{n}$ when $\textrm{Re}(\lambda_2)<\rho/2$ and of order $\log(n)\sqrt{n}$ in the critical case $\textrm{Re}(\lambda_2) = \rho/2$ . Conversely, if $\textrm{Re}(\lambda_2) > \rho/2$ , then the fluctuations are non-Gaussian and of higher order. See Janson [5] for this trichotomy and [11] for an approach based on the spectral decomposition of a suitable finite-difference transition operator on polynomial functions.

Apart from these seminal results, the model of Pólya urns has been extended and more precise asymptotics are known. Several generalizations are considered in [5]. Another possible extension is to consider measure-valued Pólya processes; see the recent work [6] for second-order asymptotics of such processes for infinitely many colors and the literature cited there for additional results.

The model and our contribution. In the current paper we consider a modification of the Pólya urn model containing two urns marked $\mathsf{U}_1$ and $\mathsf{U}_2$ . For a fixed $J\in\mathbb{N}$ representing the number of colors, we consider the random $J\times J$ matrix L as above, with independent column vectors $L^{(j)}$ and expectation matrix $A=\mathbb{E} L$ . With these initial conditions, we define the $\mathbb{N}^J$ -valued stochastic process $(B(n))_{n\in \mathbb{N}}$ as follows, with $B(n)=(B_1(n),\ldots,B_J(n))$ . Suppose that at time 0 we have one ball of type (color) $j_0$ in urn $\mathsf{U}_1$ . We draw this ball from urn $\mathsf{U}_1$ , and we put into urn $\mathsf{U}_2$ a collection of balls $L^{(j_0)}=(L^{(j_0,1)},\ldots, L^{(j_0,J)})$ (this notation means that for each $i\in [J]$ , we put $L^{(j_0,i)}$ balls of type i into urn $\mathsf{U}_2$ ). Thus we now have $\sum_{i=1}^JL^{(j_0,i)}$ balls in urn $\mathsf{U}_2$ . At the next step, we draw balls from urn $\mathsf{U}_2$ uniformly at random, one after another, and for any ball of type j drawn we add an independent collection of balls with distribution $L^{(j)}$ into urn $\mathsf{U}_1$ . We continue until urn $\mathsf{U}_2$ is empty, and then we exchange the roles of urns $\mathsf{U}_1$ and $\mathsf{U}_2$ . We emphasize that, unlike in the Pólya urn model, in the two-urn model it is more convenient to consider drawing without replacement; that is, the ball drawn in each step is not returned to either urn, but only determines the future addition of balls. In particular, all coefficients of L are nonnegative. For $j\in [J]$ , by $B_j(n)$ we denote the total number of balls of type j that have been added to one of these two urns up to (and including) the nth draw.

Graphically, we can draw a random tree in order to visualize the step-by-step evacuation of one urn and the refilling of the other one, as follows: we color the nodes of the tree in colors $\{1,\ldots,J\}$ ; the content of urn $\mathsf{U}_1$ represents the root of the tree colored with some fixed $j_0\in \{1,\ldots,J\}$ , i.e. the zero generation; after one draw of the node of type $j_0$ , the content $L^{(j_0)}$ of $\mathsf{U}_2$ represents the first generation. Then, after choosing balls (i.e. nodes of the tree at level one) uniformly at random without replacement and putting their offspring in the other urn $\mathsf{U}_1$ , we create step by step the second generation of the tree, and step by step we fill up $\mathsf{U}_1$ again. Thus what we propose here is a more refined branching process where the transition from generation k to generation $k+1$ is considered after each member of generation k reproduces. If we visualize the process as a random tree that grows after each node is chosen, the quantity $B_j(n)$ represents the number of nodes of type j and $\sum_{j=1}^JB_j(n)$ represents the total number of nodes in the tree after n steps of the process, that is, after n balls have been drawn from $\mathsf{U}_1$ and $\mathsf{U}_2$ . For better understanding, we illustrate this process in an example in Appendix A and Figure 1.

Figure 1. The model with two alternating urns and deterministic replacement matrix after $n=4$ draws.

The main focus of the current work is to investigate first- and second-order asymptotics of $B_j(n)$ as $n\to\infty$ , $j\in [J]$ . It may happen that with positive probability $L^{(j,i)}$ vanishes for all $i\in[J]$ . In such a case we do not add any new balls to the urn; we just remove the selected ball of type j. In particular, it can happen that after a finite number $n_0$ of steps, both urns are empty; in such a case we define $B(n)=B(n_0)$ , for $n\ge n_0$ . Since we are interested in the long-term behavior of the urn process, we restrict the analysis to a set where this does not happen, i.e. to the survival event $\mathcal S=\{|B(n)|\to\infty\}$ .

We can also define the corresponding sequence $(B^\circ(n))_{n\ge 0}$ that represents the types of the balls drawn up to time n. While it is possible to ask about limit theorems for $B^\circ(n)$ , the method developed in this paper for studying B(n) is directly applicable to $B^\circ(n)$ . Therefore, we focus our attention exclusively on the sequence $(B(n))_{n\in\mathbb{N}}$ .

The approach we use to investigate $(B_j(n))_{n\geq 0}$ , $j\in [J]$ , is to embed it into a multitype discrete-time branching process $(Z_n)_{n\in\mathbb{N}}$ with offspring distribution matrix L. A similar approach using the Athreya–Karlin embedding allowed Janson [5] to study $(B^\circ_j(n))_{n\geq 0}$ for the Pólya urn model. One difference between our model and the one in [5] is that, in the latter, the process is embedded into a multitype continuous-time Markov branching process, and an individual reproduces after an exponential clock rings. In our model, in the embedded branching process, an individual reproduces after deterministic time 1. The lattice nature of the model manifests itself in the second-order behavior of $(B_{j}(n))_{n\geq 0}$ .

Assumptions. In this work we use the following assumptions:

1. (GW1) The matrix A has spectral radius $\rho>1$ .

2. (GW2) The matrix A is positively regular.

3. (GW3) We have $\textbf{0}\neq \sum_{j=1}^J\textrm{Cov}\left[L^{(j)}\right]$ and $\textrm{Var}[L^{(i,j)}]<\infty$ for all $i,j\in[J]$ .

4. (GW4) For every $i,j\in[J]$ , the expectation $\mathbb{E}[ L^{(i,j)}\log L^{(i,j)}]$ is finite.

In the third condition above, $\textbf{0}$ is the $J\times J$ zero matrix, and for $j\in[J], \textrm{Cov}\left[L^{(j)}\right]$ represents the $J \times J$ covariance matrix of the vector $L^{(j)}$ . If the matrix A is irreducible, the Perron–Frobenius theorem ensures that the dominant eigenvalue $\rho$ of A is real, positive, and simple. If $\rho>1$ , this means that the multitype branching process with offspring distribution matrix L is supercritical, i.e. $\mathbb{P}(\mathcal S)>0$ . If $\textrm u$ is the corresponding eigenvector for $\rho$ , then clearly all the entries $\textrm u_j$ are strictly greater than zero for any $j\in[J]$ , and we assume that $\textrm u$ is normalized in such a way that $\sum_{j\in[J]}\textrm u_j=1$ . First-order asymptotics of $B_j(n)$ , $j\in[J]$ , are determined by $\rho$ and the vector $\textrm u$ .

Theorem 1. Assume (GW1), (GW2), and (GW4) hold. Then for any $j\in [J]$ we have the following strong law of large numbers for the total number of balls of type j after n draws:

\begin{align*} \lim_{n\to\infty}\frac{B_j(n)}{n}=\rho \textrm u_j,\quad \quad \mathbb{P}^\mathcal{S}\textit{-almost surely}. \end{align*}

Thus, the first-order behavior of $B_j(n)$ , $j\in [J]$ resembles the first-order behavior of $B^\circ_j(n)$ from the model with one urn [5]. In order to understand the second-order asymptotics of $B_j(n)$ , we need full information on the spectral decomposition of the mean replacement matrix A. We denote by $\sigma_A$ the spectrum of the matrix A and define $\sigma_A^1=\{\lambda\in\sigma_A\;:\;\ |\lambda|>\sqrt{\rho}\}$ , $\sigma_A^2=\{\lambda\in\sigma_A\;:\;\ |\lambda|=\sqrt{\rho}\}$ , and finally $\sigma_A^3=\{\lambda\in\sigma_A\;:\;\ |\lambda|<\sqrt{\rho}\}$ . Then we can write

\begin{align*}\sigma_A=\sigma_A^{1}\cup\sigma_A^{2}\cup\sigma_A^{3}.\end{align*}

For a simple eigenvalue $\lambda\in\sigma_A$ , we denote by $\textrm u^\lambda$ and $\textrm v^\lambda$ the corresponding left and right eigenvectors. We set

\begin{align*}\gamma=\max\{|\lambda|\;:\;\lambda\in\sigma_A\setminus\{\rho\}\}\quad \text{and}\quad \Gamma=\{\lambda\in\sigma_A\;:\;|\lambda|=\gamma\},\end{align*}

so $\rho-\gamma$ is the spectral gap of A, and $\Gamma$ is the set of eigenvalues of A where the spectral gap is achieved. For a complex number z we set $\log_\rho z=\frac{\log z}{\log \rho},$ where $z\mapsto\log z$ is the principal branch of the natural logarithm. Denote by $\{\textrm e_j\}_{j\in[J]}$ the standard basis vectors in $\mathbb{R}^J$ . For $j\in [J]$ , consider the following row vector in $\mathbb{R}^{1\times J}$ :

(1) \begin{equation} \textrm{w}_j=\textrm{e}_j^{\top}A-\rho\textrm{u}_j\textrm{1}, \end{equation}

where the ith entry is $\textrm{w}_{ji}=a_{ji}-\rho\textrm{u}_j=\mathbb{E}[L^{(i,j)}]-\rho\textrm{u}_j$ , for $1\leq i\leq J$ . In the above equation, 1 denotes the vector in $\mathbb{R}^{1\times J}$ with all entries equal to one. Our main result provides the second-order behavior of $(B_j(n))_{n\in \mathbb{N}}$ .

Theorem 2. Assume that (GW1)–(GW3) hold, all $\lambda\in\Gamma$ are simple, and there exists $\delta>0$ such that $\mathbb{E}\big[\big(L^{(i,j)}\big)^{2+\delta}\big]<\infty$ for all $i,j\in[J]$ . Then for any $j\in[J]$ the following trichotomy holds:

1. (i) If $\gamma>\sqrt \rho$ , then for any $\lambda\in\Gamma$ there exist a 1-periodic, continuous function $f_\lambda\;:\;\mathbb{R}\to\mathbb{C}$ and random variables $X,X_\lambda$ such that the following holds:

   \begin{align*}B_j(n)=\rho \textrm u_j \cdot n+\sum_{\lambda\in \Gamma}n^{\log_\rho\lambda}f_\lambda(\log_\rho n-X)X_\lambda+o_{\mathbb{P}}\big(n^{\log_\rho\gamma}\big).\end{align*}

2. (ii) If $\gamma = \sqrt \rho$ and for some $\lambda\in\sigma^{2}_A$ and $i\in[J]$ we have $\textrm{w}_j\textrm u^\lambda\neq0$ and $\textrm{Var}[\textrm v^\lambda L\textrm{e}_i]>0$ for $\textrm{w}_j$ defined as in (1), then there exist a 1-periodic, continuous function $\psi\;:\;\mathbb{R}\to(0,\infty)$ and a random variable X such that, conditionally on $\mathcal{S}$ , the following convergence in distribution holds:

   \begin{align*}\frac{B_j(n)-\rho \textrm u_j\cdot n}{\sqrt {n\log_{\rho} n}\ \psi(\log_\rho n-X)}\xrightarrow{\scriptscriptstyle{\textrm{d}}} \mathcal{N}(0,1),\qquad \text{as }n\to\infty.\end{align*}

3. (iii) If $\gamma < \sqrt \rho$ and for some $\lambda\in\sigma^{3}_A$ and some $i\in[J]$ we have $\textrm{w}_j\textrm u^\lambda\neq0$ and $\textrm{Var}[\textrm v^\lambda L\textrm{e}_i]>0$ with $\textrm{w}_j$ defined as in (1), then there exist a 1-periodic, continuous function $\psi\;:\;\mathbb{R}\to(0,\infty)$ and a random variable X such that, conditionally on $\mathcal{S}$ , the following convergence in distribution holds:

   \begin{align*}\frac{B_j(n)-\rho \textrm u_j\cdot n}{\sqrt n\ \psi(\log_\rho n-X)}\xrightarrow{\scriptscriptstyle{\textrm{d}}} \mathcal{N}(0,1),\qquad \text{as }n\to\infty.\end{align*}

A more general and quantified result where the periodic functions are explicitly defined is provided in Theorem 4.

Note that the result above slightly differs from that of the one-urn model, where the functions $\psi$ and $f_\lambda$ are actually constants. What might be even more surprising is that, in our model, the phase transition occurs at $\sqrt{\rho}$ rather than at $\rho/2$ as observed in the Pólya urn model. The heuristic explanation is as follows: the growth in mean of the corresponding continuous-time branching process is driven by the semigroup $e^{tA}$ . In particular, the leading asymptotic is $e^{t\rho}$ and the next order is $|e^{t \lambda_2}|=e^{t \textrm{Re}\lambda_2}$ . We anticipate observing Gaussian fluctuations in the branching process at the scale $\sqrt{e^{t\rho}}$ (with possible polynomial corrections), providing a natural threshold for $\textrm{Re}\lambda_2$ in relation to $\rho/2$ . On the other hand, the two-urn model is embedded into a discrete-time branching process whose growth in the mean is driven by the semigroup $A^n$ (or $(I+A)^n$ in the model with no replacement). Thus, the leading term is at scale $\rho^n$ and the subleading term is at scale $|\lambda_2|^n$ . As before, we expect to observe Gaussian fluctuations at scale $\sqrt{\rho^n}$ , which induce natural distinctions depending on the relative locations of $|\lambda_2|$ and $\sqrt{\rho}$ .

Structure of the paper. In Section 2 we introduce multitype branching processes $(Z_n)_{n\in\mathbb{N}}$ and Crump–Mode–Jagers processes $(Z_n^{\Phi})_{n\in\mathbb{N}}$ counted with a characteristic $\Phi\;:\;\mathbb{Z}\to \mathbb{R}^{1\times J}$ . Then in Section 3 we show how to relate our model $(B_j(n))$ , with two interacting urns, to a branching process $(Z^{\Phi^j}_n)_{n\in \mathbb{N}}$ counted with a characteristic $\Phi^j$ . By applying [8, Proposition 4.1] to $(Z^{\Phi^j}_n)_{n\in\mathbb{N}}$ , we then obtain first-order asymptotics of $(B_j(n))_{n\in\mathbb{N}}$ for any $j\in[J]$ (Theorem 1). The main result is proved in Section 4. We conclude with Appendix A, where the model with two interacting urns is illustrated by an example with deterministic replacement matrix, and Appendix B, where higher-moment estimates for $(Z_n^{\Phi})$ are given for general characteristics $\Phi$ .

2. Preliminaries

For the rest of the paper, $(\Omega,\mathcal{A},\mathbb{P})$ is a probability space on which all the random variables and processes we consider are defined. We write $\xrightarrow{\scriptscriptstyle{a.s.}}$ for almost sure convergence, $\xrightarrow{\scriptscriptstyle{\mathbb{P}}}$ for convergence in probability, $\xrightarrow{\scriptscriptstyle{\textrm{d}}}$ for convergence in distribution, and $\xrightarrow{\scriptscriptstyle{\textrm{st}}}$ for stable convergence (cf. [1] for the definition and properties). We also use $\mathbb{P}^\mathcal{S}$ to denote the conditional probability $\mathbb{P}[\cdot|\mathcal{S}]$ , and the corresponding convergences are denoted by $\xrightarrow{\scriptscriptstyle{\mathbb{P}^\mathcal{S}}},\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}},\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}}$ . We use the notation $\mathbb{N} =\{1,2,\ldots,\}$ and $\mathbb{N}_0 =\{0,1,2,\ldots,\}$ .

Stochastic processes. Our convergence results for stochastic processes use the usual space $\mathcal{D}$ of right-continuous functions with left-hand limits, always equipped with the Skorokhod $\textbf{J}_1$ topology. For a finite-dimensional vector space E and any interval $J\subseteq [-\infty,\infty]$ , we denote by $\mathcal{D}(J)=\mathcal{D}(J,E)$ the space of all right-continuous functions from J to E with left-hand limits.

For an $n\times m$ matrix $A=(a_{ij})_{i,j}$ with $m,n\in\mathbb{N}$ , the Hilbert–Schmidt norm of A, also called the Frobenius norm, is defined as

\begin{align*}\|A\|_{HS}=\Big(\sum_{i=1}^n\sum_{j=1} ^m|a_{ij}|^2\Big)^{1/2}.\end{align*}

Since for any vector its Hilbert–Schmidt norm coincides with its Euclidean norm, for the rest of the paper we write only $\|\cdot\|$ instead of $\|\cdot\|_{HS}$ .

Ulam–Harris tree $\mathcal{U}_{\infty}$ . An Ulam–Harris tree $\mathcal{U}_{\infty}$ is an infinite rooted tree with vertex set $V_{\infty}= \bigcup_{n \in \mathbb{N}_0} \mathbb{N}^n$ , the set of all finite strings or words $i_1\cdots i_n$ of positive integers over n letters, including the empty word $\varnothing$ (which we take to be the root), and with an edge joining $i_1\cdots i_n$ and $i_1\cdots i_{n+1}$ for any $n\in \mathbb{N}_0$ and any $i_1, \cdots, i_{n+1}\in \mathbb{N}$ . Thus every vertex $v=i_1\cdots i_n$ has outdegree $\infty$ , and the children of v are the words $v1,v2,\ldots$ . We let them have this order so that $\mathcal{U}_{\infty}$ becomes an infinite ordered rooted tree. We will identify $\mathcal{U}_{\infty}$ with its vertex set $V_{\infty}$ , when there is no risk of confusion in doing so. For vertices $v=i_1\cdots i_n$ we also write $v=(i_1,\ldots,i_n)$ , and if $u=(j_1,\ldots,j_m)$ we write uv for the concatenation of the words u and v; that is, $uv=(j_1,\ldots,j_m,i_1,\ldots,i_n)$ . The parent of $i_1\cdots i_n$ is $i_1\cdots i_{n-1}$ . Finally, for $u \in \mathcal{U}_{\infty}$ , we use the notation $|u|=n$ to mean $u \in\mathbb{N}^n$ (i.e. u is a word of length (or height) n; in other words, it is at distance n from the root $\varnothing$ ). For any $u\in V_{\infty}$ , by $\mathbb{T}_u$ we mean the subtree of $\mathcal{U}_{\infty}$ rooted at u, that is, u together with all infinite paths going away from u, and for $u,v\in \mathcal{U}_{\infty}$ we denote by d(u, v) their graph distance. For trees rooted at $\varnothing$ , we omit the root and we write only $\mathbb{T}$ . For $J\in \mathbb{N}$ , a J-type tree is a pair $(\mathbb{T},\textrm t)$ where $\mathbb{T}$ is a rooted subtree of $\mathcal{U}_{\infty}$ and $\textrm t\;:\;\mathbb{T}\to\{1,\ldots,J\}$ is a function defined on the vertices of $\mathbb{T}$ that returns for each vertex v its type $\textrm t(v)$ .

Multitype branching processes. Consider the random $J\times J$ matrix L with independent column vectors $L^{(j)}$ , for $1\leq j\leq J$ , as in the introduction. Multitype Galton–Watson trees are random variables taking values in the set of J-type trees $(\mathbb{T},\textrm t)$ , where the type function $\textrm t$ is random and defined in terms of the matrix L. Let $(L(u))_{u\in \mathcal{U}_{\infty}}$ be a family of independent and identically distributed (i.i.d.) copies of L indexed over the vertices of $\mathcal{U}_{\infty}$ . For any $i\in [J]$ , we define recursively the random labeled tree $\mathbb{T}^{i}$ rooted at $\varnothing$ , with the associated type function $\textrm t=\textrm t^{i}\;:\;\mathbb{T}^{i}\to \{1,\ldots,J\}$ , as follows: $\varnothing\in\mathbb{T}^{i}$ and $\textrm t(\varnothing)=i$ . Now suppose that $u=j_1\ldots j_m\in\mathbb{T}^{i}$ with $\textrm t(u)=j$ , for some $j\in[J]$ . Then

\begin{align*}j_1 \ldots j_m k\in \mathbb{T}^{i} \quad\text{if and only if}\quad k\le L^{(j,1)}(u)+\cdots+L^{(j,J)}(u),\end{align*}

and we set $\textrm{t}(u k)=\ell$ whenever

\begin{align*}L^{(j,1)}(u)+\cdots+L^{(j,\ell-1)}(u)<k\le L^{(j,1)}(u)+\cdots+L^{(j,\ell)}(u).\end{align*}

The multitype branching process $Z_n=(Z^1_n, \ldots , Z^J_n)$ associated with the pair $(\mathbb{T}^{i_0},\textrm t)$ , and starting from a single particle (or individual) of type $i_0\in [ J]$ at the root $\varnothing$ (i.e., $\textrm t(\varnothing)=i_0$ ) is defined as follows: $Z_0=\textrm{e}_{i_0} $ , and for $n\geq 1$ ,

\begin{align*}Z^i_n=\#\{u\in\mathbb{T}^{i_0}\;:\;|u|=n\text{ and }\textrm t(u)=i\}, \quad \text{for } i\in [J],\end{align*}

so $Z_n^i$ represents the number of individuals of type i in the nth generation, or the number of vertices $u\in \mathbb{T}^{i_0}$ with $|u|=n$ and $\textrm t(u)=i$ . The main results of [8] that we use in the current paper hold under the assumptions (GW1)–(GW3) on $(Z_n)_{n\in \mathbb{N}}$ , which we suppose to hold here as well. In particular, $(Z_n)_{n\in\mathbb{N}}$ is a supercritical branching process.

Spectral decomposition of A . Recall the decomposition of the spectrum $\sigma_A$ of the matrix A as $\sigma_A=\sigma_A^{1}\cup\sigma_A^{2}\cup\sigma_A^{3}$ . From the Jordan–Chevalley decomposition of A (which is unique up to the order of the Jordan blocks) we infer the existence of projections $(\pi_\lambda)_{\lambda\in \sigma_A}$ that commute with A and satisfy $\sum_{\lambda\in\sigma_A}\pi_{\lambda}=I$ and

\begin{align*}A\pi_{\lambda}=\pi_{\lambda}A=\lambda\pi_{\lambda}+N_{\lambda},\end{align*}

where $N_{\lambda}=\pi_{\lambda}N_{\lambda}=N_{\lambda}\pi_{\lambda}$ is a nilpotent matrix. Moreover, for any $\lambda_1,\lambda_2\in\sigma_A$ it holds that $\pi_{\lambda_1}\pi_{\lambda_2}=\pi_{\lambda_1}\textbf{1}_{\{\lambda_1=\lambda_2\}}$ . If $\lambda\in\sigma_A$ is a simple eigenvalue of A and $\textrm u^\lambda, \textrm v^\lambda$ are the corresponding left and right eigenvectors, normalized in such a way that $\textrm v^\lambda\textrm u^\lambda=1$ , then $\pi_\lambda=\textrm u^\lambda\textrm v^\lambda$ . If we write $N=\sum_{\lambda\in\sigma_A}N_{\lambda}$ , then N is also a nilpotent matrix and we have $N\pi_{\lambda}=N_{\lambda}$ . Thus A can be decomposed into its semisimple part $D=\sum_{\lambda\in\sigma_A}\lambda\pi_{\lambda}$ and a nilpotent part N, as $A=D+N$ .

For any $\lambda\in\sigma_A$ , we denote by $d_{\lambda}\geq 0$ the integer such that $N_{\lambda}^{d_{\lambda}}\neq 0$ but $N_{\lambda}^{d_{\lambda}+1}=0$ (hence $d_{\lambda}+1$ is at most the multiplicity of $\lambda$ ). So $d_{\lambda}=0$ if and only if $N_{\lambda}=0$ , and this happens for all $\lambda$ if and only if A is diagonalizable (that is, A has a complete set of J independent eigenvectors). Since $\rho$ is a simple eigenvalue, we have $N_{\rho}=0$ and $d_{\rho}=0$ , and $\pi_{\rho}=\textrm u\textrm v$ . We set

\begin{align*}\pi^{(1)}=\sum_{\lambda\in\sigma^1_A}\pi_\lambda,\quad\pi^{(2)}=\sum_{\lambda\in\sigma^2_A}\pi_\lambda,\quad\pi^{(3)}=\sum_{\lambda\in\sigma^3_A}\pi_\lambda,\end{align*}

and for $i=1,2$ , we define

\begin{align*}A_i= A\pi^{(i)}+\big(I-\pi^{(i)}\big).\end{align*}

The process $\big(W_n^{(i)}\big)_{n\in \mathbb{N}}$ defined by

\begin{align*}W_n^{(i)}= A_i^{-n}\pi^{(i)} Z_n\end{align*}

is a $\mathcal{A}_n$ -martingale, where $(\mathcal{A}_n)_{n\ge0}$ is the filtration $\mathcal{A}_n=\sigma(\{L(u)\;:\;|u|\le n\})$ . According to [8, Lemma 2.2], $W_n^{(1)}$ converges in $\mathcal{L}^2(\Omega,\mathcal{A},\mathbb{P})$ to a random variable $W^{(1)}$ whose expectation is $\mathbb{E} W^{(1)}=\pi^{(1)}\textrm e_{i_0}$ . In particular, we have the convergence

\begin{align*}\rho^{-n} Z_n\to \pi_\rho W^{(1)} = W \textrm u,\quad \text{as } n\to\infty,\end{align*}

in $\mathcal{L}^2$ , and the random variable W is given by $W= \textrm v \cdot W^{(1)}$ with $\mathbb{E} W=\textrm v^{i_0}>0$ . The classical Kesten–Stigum theorem [7] states that under (GW1), (GW2), and (GW4) the convergence above holds almost surely. For the rest of the paper, when we use the random variable W, we always mean the limit random variable from the Kesten–Stigum theorem.

2.1. Branching processes counted with a characteristic

We recall that a characteristic of dimension one is a function $\Phi\;:\;\mathbb{Z}\to\mathbb{R}^{1\times J}$ , so that for each $k\in \mathbb{Z}$ , $\Phi(k)$ is an $\mathbb{R}^{1\times J}$ -valued random variable defined on the same probability space $(\Omega,\mathcal{A},\mathbb{P})$ where the Galton–Watson process $(Z_n)_{n\in\mathbb{N}}$ and its genealogical tree $\mathbb{T}$ are defined. A deterministic characteristic is just a fixed function $\Phi\;:\;\mathbb{Z}\to\mathbb{R}^{1\times J}$ . For a random function $\Phi\;:\;\mathbb{Z}\to \mathbb{R}^{1\times J}$ and the multitype Galton–Watson tree $\mathbb{T}$ , the process $(Z^\Phi_n)_{n\in \mathbb{N}}$ which for any $n\in\mathbb{N}$ is defined as

\begin{align*}Z^\Phi_n=\sum_{u\in\mathbb{T}}\Phi_u(n-|u|)\textrm{e}_{\textrm t(u)}\end{align*}

is called the multitype Crump–Mode–Jagers (CMJ) process counted with characteristic $\Phi$ , or simply the branching process counted with characteristic $\Phi$ , where $\left(\Phi_u\right)_{u\in\mathcal{U}_{\infty}}$ is an i.i.d. copy of $\Phi$ . First- and second-order asymptotics for $(Z^\Phi_n)_{n\in \mathbb{N}}$ , under mild assumptions on $\Phi$ , are considered in [8, Proposition 4.1] and in [8, Theorem 3.5], respectively. We use these two results below for a particular choice of the characteristic $\Phi$ , and show how the branching process with this particular choice of characteristic can be related to the two-urn model with alternating urns $\mathsf{U}_1$ and $\mathsf{U}_2$ .

The choice of the characteristic. Let $U\sim\mathsf{Unif}[0,1]$ be a uniform random variable taking values in [0,1] and defined on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ . For every threshold $x\in[0,1)$ we define the characteristic $\Phi^{\mathsf{t}}_{x}\;:\;\mathbb{N}_0\to\mathbb{R}^{1\times J}$ by

(2) \begin{align} \Phi_x^{\mathsf{t}}(k)\textrm{e}_i=\textbf{1}_{\{{k\geq 1}\}}+\textbf{1}_{\{k=0\}}\textbf{1}_{\{U \le x\}}, \quad \text{for } 1\leq i\leq J,\end{align}

or, in a simplified way, for $1\leq i\leq J$ ,

\begin{equation*}\Phi_x^{\mathsf{t}}(k)\textrm{e}_i=\begin{cases}1,&\quad \text{for } k\geq 1,\\[5pt] 1 \text{ with probability } \mathbb{P}(U\leq x)=x,& \quad \text{for } k=0, \\[5pt] 0 \text{ with probability } \mathbb{P}(U> x)=1-x,& \quad \text{for } k=0.\end{cases}\end{equation*}

Similarly, for $j\in[J]$ , we define $\Phi_{x}^j\;:\;\mathbb{N}_0\to\mathbb{R}^{1\times J}$ by

(3) \begin{align}\Phi_x^j(k)\textrm{e}_i= \textbf{1}_{\{k\geq 1\}}\underbrace{\langle\textrm{e}_j,L\textrm{e}_i\rangle}_{=L^{(i,j)}}+\textbf{1}_{\{k=0\}}\textbf{1}_{\{U \le x\}}\langle\textrm{e}_j,L\textrm{e}_i\rangle, \quad \text{for } 1\leq i\leq J,\end{align}

so $\Phi_x^j(k)\textrm{e}_i=\Phi^{\mathsf{t}}_x(k)\textrm{e}_i L^{(i,j)}$ . For the uniform random variable U on [0,1], let $(U_u)_{u\in\mathcal{U}_{\infty}}$ be an i.i.d. copy of U. For $u\in \mathcal{U}_{\infty}$ , let $ \Phi^{\mathsf{t}}_{x,u}$ (respectively $ \Phi^j_{x,u}$ ) be defined through (2) (respectively (3)) with U, L being replaced by $U_u,\ L(u)$ . Then the family $\big((L(u),\Phi^{\mathsf{t}}_{x,u},\Phi_{x,u}^j)\big)_{u\in \mathcal{U}_{\infty}}$ is an i.i.d. collection of copies of $(L,\Phi^{\mathsf{t}}_x,\Phi_x^j)$ , for $j\in [J]$ .

For the characteristic $\Phi_x^{\mathsf{t}}$ from (2), threshold $x\in [0,1)$ , and multitype Galton–Watson tree $\mathbb{T}$ , the process $\big(Z^{\Phi_x^{\mathsf{t}}}_n\big)_{n\geq 0}$ counts the total number of individuals $u\in \mathcal{U}_{\infty}$ born before time n (including the root), and those born at time n with $U_u\le x$ , since we have

\begin{align*}Z_n^{\Phi^{\mathsf{t}}_x}&=\sum_{u\in \mathbb{T}}\Phi^{\mathsf{t}}_{x,u}(n-|u|)\textrm{e}_{\textrm t(u)}=\sum_{k=0}^{n-1}\sum_{u\in \mathbb{T}; |u|=k}1+\sum_{u\in \mathbb{T}; \ |u|=n}\textbf{1}_{\{U_u\leq x\}}\\[5pt] &=\sum_{k=0}^{n-1}|Z_k|+\sum_{u\in \mathbb{T}; \ |u|=n}\textbf{1}_{\{U_u\leq x\}},\end{align*}

where $|Z_n|=\sum_{j=1}^JZ_n^j$ represents the size of the nth generation of the Galton–Watson process. On the other hand, $L\textrm e_i=L^{(i)}$ represents the random collection of individuals born from an individual of type i, while $L^{(i,j)}=\langle\textrm{e}_j,L\textrm{e}_i\rangle$ represents the random number of offspring of type j of an individual of type i for $i,j\in[J]$ . Therefore $Z_n^{\Phi_x^j}$ counts the number of individuals of type j born up to time n, and those of type j born at time $n+1$ but with threshold $\le x$ , so $Z_n^{\Phi^j_x}$ can be written as

\begin{align*}Z_n^{\Phi^j_x}&=\sum_{k=0}^{n-1}\sum_{u\in\mathbb{T}; |u|=k}\langle\textrm{e}_j,L(u)\textrm{e}_{\textrm t(u)}\rangle+\sum_{u\in \mathbb{T};|u|=n+1,\textrm t(u)=j}\textbf{1}_{\{U_u\leq x\}},\end{align*}

since $Z_{k+1}^j=\sum_{u\in \mathbb{T};|u|=k}\langle\textrm{e}_j,L(u)\textrm{e}_{\textrm t(u)}\rangle$ represents the number of offspring of type j in the $(k+1)$ th generation and $|\{u\in \mathbb{T};\;|u|=n+1,\textrm t(u)=j\}|=Z_{n+1}^j$ . Summing up over all $j\in [J]$ gives

\begin{align*}Z_{n+1}^{\Phi^{\mathsf{t}}_x}-1= \sum_{j=1}^J Z_n^{\Phi^j_x},\end{align*}

and an application of [8, Proposition 4.1] to $Z_n^{\Phi^{\mathsf{t}}_x}$ and $Z_n^{\Phi^j_x}$ , for $j\in[J]$ , yields the law of large numbers.

Proposition 1. Under the assumptions (GW1), (GW2), and (GW4), for any threshold $x\in[0,1)$ and characteristic $\Phi_x^{\mathsf{t}}$ (respectively $\Phi_x^j$ , $j\in[J]$ ) defined in (2) (respectively (3)) we have the following:

\begin{align*}\lim_{n\to\infty}\dfrac{Z_n^{\Phi^{\mathsf{t}}_x}}{\rho^n}&=\Big(\frac{1}{\rho-1}+x\Big)W,\quad \text{$\mathbb{P}^\mathcal{S}$}\textit{-almost\;surely},\\[5pt] \lim_{n\to\infty}\dfrac{Z_n^{\Phi^j_x}}{\rho^n}&=\Big(\frac{1}{\rho-1}+x\Big)W\rho\textrm u_j,\quad \text{$\mathbb{P}^\mathcal{S}$}\textit{-almost surely}.\end{align*}

Proof. Since for any fixed $x\in[0,1)$ the random variables $(\textbf{1}_{\{U_u\leq x\}})_{u\in\mathcal{U}_{\infty}}$ are i.i.d. and Bernoulli-distributed as $\mathsf{Bern}(x)$ , in view of the strong law of large numbers we obtain

\begin{align*}\lim_{n\to\infty}\frac{1}{|Z_n|} \sum_{u\in \mathbb{T}; \ |u|=n}\textbf{1}_{\{U_u\leq x\}}=x,\quad \text{$\mathbb{P}^\mathcal{S}$-almost surely},\end{align*}

and similarly

\begin{align*}\lim_{n\to\infty}\frac{1}{Z_n^j} \sum_{|u|=n,\textrm t(u)=j}\textbf{1}_{\{U_u\leq x\}}=x,\quad \text{$\mathbb{P}^\mathcal{S}$-almost surely}.\end{align*}

For the deterministic characteristic $\textbf{1}_{\{k\geq 1\}}$ and the corresponding branching process counted with this characteristic, by applying [8, Proposition 4.1(i)] we get

\begin{align*}\frac{Z_n^{\Phi_x^{\mathsf{t}}}}{\rho^{n}}&=\frac{1}{\rho}\cdot\overbrace{\frac{1}{\rho^{n-1}}\sum_{k\geq 0}^{n-1}\sum_{|u|=k}1}^{\xrightarrow{\scriptscriptstyle{a.s.}} W\sum_{k\geq 0}\rho^{-k}}+\overbrace{\frac{1}{\rho^{n}}\sum_{|u|=n}\textbf{1}_{\{U_u\leq x\}}}^{\xrightarrow{\scriptscriptstyle{a.s.}} xW}\\[5pt] &\xrightarrow{\scriptscriptstyle{a.s.}} \Big(\frac{1}{\rho}\sum_{k\geq 0}\frac{1}{\rho^{k}}+x\Big)W=\Big(\frac{1}{\rho-1}+x\Big)W,\quad \text{ as }n\to\infty, \end{align*}

and this shows the first part of the claim. For the second one, from the Kesten–Stigum theorem we know that $\frac{Z_n^j}{\rho^n}\xrightarrow{\scriptscriptstyle{a.s.}} W\textrm{u}_j$ as $n\to\infty$ for any $j\in[J]$ , and for the characteristic $\Phi_x^j$ , since we have

\begin{align*}Z_n^{\Phi_x^j}=\sum_{k=0}^{n}Z_k^j+\sum_{u\in \mathbb{T}; \ |u|=n+1,\textrm t(u)=j}\textbf{1}_{\{U_u\leq x\}},\end{align*}

we obtain

\begin{align*}\frac{Z_n^{\Phi_x^j}}{\rho^{n}}&=\overbrace{\frac{1}{\rho^{n}}\sum_{k=1}^{n}Z_k^j}^{\xrightarrow{\scriptscriptstyle{a.s.}} W\sum_{k\geq 0}\rho^{-k}\textrm{u}_j}+\overbrace{\rho\frac{1}{\rho^{n+1}}\sum_{|u|=n+1,\textrm t(u)=j}\textbf{1}_{\{U_u\leq x\}}}^{\xrightarrow{\scriptscriptstyle{a.s.}} \rho xW \textrm{u}_j}\\[5pt] &\xrightarrow{\scriptscriptstyle{a.s.}} \Big(\frac{1}{\rho}\sum_{k\geq 0}\frac{1}{\rho^{k}}+x\Big)W\rho \textrm{u}_j=\Big(\frac{1}{\rho-1}+x\Big)W\rho \textrm{u}_j,\quad \textit{ as }n\to\infty, \end{align*}

and the proof is completed.

Following the notation from [8], for every characteristic $\Phi\;:\;\mathbb{Z}\to\mathbb{R}^{1\times J}$ we define the two vectors $\textrm{x}_i(\Phi)=\sum_{k\in \mathbb{N}}\mathbb{E}[\Phi(k)]\pi^{(i)}A_i^{-k}$ , for $i=1,2$ . In particular, since $\mathbb{E}[\Phi_x^{\mathsf{t}}(0)]=x\textrm{1}$ , we have

(4) \begin{align} \nonumber\textrm{x}_i\left(\Phi_x^{\mathsf{t}}\right)&=x\textrm{1}\pi^{(i)}+\textrm{1}\pi^{(i)}\sum_{k=1}^{\infty}A_i^{-k}=\textrm{1}\pi^{(i)}\left(xI+\sum_{k=1}^{\infty}A_i^{-k}\right)\\[5pt] &=\textrm{1}\pi^{(i)}\left(x + (A_i-I)^{-1}\right),\quad \text{for }i=1,2.\end{align}

For any random characteristic $\Phi\;:\;\mathbb{Z}\to\mathbb{R}^{1\times J}$ that satisfies the assumptions of [8, Theorem 3.5]—i.e. for which (GW1)–(GW3) hold, $\sum_{k\in \mathbb{Z}}\big\|\mathbb{E}[\Phi(k)]\big\|(\rho^{-k}+\vartheta^{-k})<\infty$ for some $\vartheta<\sqrt{\rho}$ , and finally $\sum_{k\in \mathbb{Z}}\|\textrm{Var}[\Phi(k)]\|\rho^{-k}<\infty$ —we set

(5) \begin{equation}F^\Phi_n=\textrm{x}_1(\Phi)A^n_1 W^{(1)}+\textrm{x}_2(\Phi)A^n_2Z_0.\end{equation}

Recall that the constants $\sigma_{\ell}^2$ , for $\ell=0,\ldots, J-1$ , are defined as

(6) \begin{equation}\sigma_{\ell}^2=\sigma_{\ell}^2(\Phi)= \frac{\rho^{-\ell}}{(2\ell+1)(\ell !)^2}\sum_{\lambda\in\sigma^2_A}\textrm{Var} \Big[\textrm{x} _2(\Phi) \pi_\lambda (A-\lambda I)^{\ell}L\Big]\textrm{u}.\end{equation}

Let $\ell$ be the maximal integer such that $\sigma_{\ell}(\Phi)>0$ , and set $\ell=-\frac12$ if there is no such integer. Then Theorem 3.5 of [8] states that, for a standard normal variable $\mathcal{N}(0,1)$ independent of W, the following stable convergence holds:

(7) \begin{align}\frac{Z^\Phi_n-F^\Phi_n}{n^{\ell+\frac12}\rho^{n/2}\sqrt{W}}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} G^{\Phi}, \quad \text{as } n\to\infty,\end{align}

where $G^\Phi= \sigma_{\ell}(\Phi) \mathcal{N}(0,1)$ if not all $\sigma_{\ell}$ are zero, and $G^\Phi= \sigma(\Phi)\mathcal{N}(0,1)$ otherwise, while $\sigma(\Phi)$ is defined by

(8) \begin{align} \sigma^2(\Phi)=\sum_{k\in \mathbb{Z}}\rho^{-k}\textrm{Var}\left[\Phi(k)+\Psi^\Phi(k)\right]\textrm{u},\end{align}

and $\Psi^\Phi$ is the centered characteristic given by

\begin{align*} \Psi^\Phi(k)= \sum_{\ell\in \mathbb{Z}}\mathbb{E}\Phi(k-\ell-1)A^{\ell}\mathsf{P}(k,\ell)(L-A).\end{align*}

Above, the matrices $\mathsf{P}(k, \ell)$ , for $k,\ell\in\mathbb{Z}$ are defined as

\begin{align*} \mathsf{P}(k,\ell)=\begin{cases} -\pi^{(1)}\textbf{1}_{\{\ell<0\}}+\pi^{(2)}\textbf{1}_{\{\ell\geq 0\}}+\pi^{(3)}\textbf{1}_{\{\ell\geq 0\}},\quad \text{if }k\leq 0,\\[5pt] -\pi^{(1)}\textbf{1}_{\{\ell<0\}}-\pi^{(2)}\textbf{1}_{\{\ell<0\}}+\pi^{(3)}\textbf{1}_{\{\ell\geq 0\}},\quad \text{if }k>0. \end{cases}\end{align*}

3. The embedding of the urn model into the branching process

Notation. We slightly abuse notation and write $\Phi^{\mathsf{t}}$ (respectively $\Phi^j$ , $j\in[J]$ ) for the whole family $\big\{\Phi^{\mathsf{t}}_x\big\}_{x\in[0,1)}$ (respectively $\big\{\Phi^j_x\big\}_{x\in[0,1)}$ ) of characteristics indexed over the threshold $x\in[0,1)$ . We denote by $\mathcal C$ the set of characteristics $\Phi$ which are linear combinations of $\Phi^{\mathsf{t}}$ and $\Phi^j$ , for $j\in[J]$ . Again by abuse of notation, by $\Phi\in\mathcal{C}$ we actually refer to the whole family of characteristics $\{\Phi_x\}_{x\in[0,1)}$ ; that is,

\begin{align*}\Phi=\big\{\Phi_x\in\mathcal{C}\;:\; x\in[0,1)\big\}=\big\{a \Phi^{\mathsf{t}}_x+b \Phi^j_x\;:\; a,b\in\mathbb{R},\ j\in[J], \ x\in [0,1)\big\}.\end{align*}

Extension of $x\in[0,1)$ to $x\in\mathbb{R}$ . Instead of working with thresholds $x\in[0,1)$ and corresponding characteristics $\Phi_x$ , we can extend the domain of x to the whole of $\mathbb{R}$ as follows. For any $\Phi\in\mathcal C$ and any $x\in\mathbb{R}$ we define

\begin{align*}\Phi_{x}(k)=\Phi_{\{x\}}(k+\lfloor{x}\rfloor).\end{align*}

The corresponding CMJ process $(Z_n^{\Phi_x})_{n\in\mathbb{N}}$ satisfies $Z_n^{\Phi_x}=Z_0^{\Phi_{x+n}}$ for every $n\in\mathbb{N}$ and $x\in\mathbb{R}$ , and finally we define

\begin{align*}\mathcal{Z}^{\Phi}(x)= Z_0^{\Phi_{x}},\end{align*}

and similarly $\mathcal{F}^{\Phi}(x)= F_0^{\Phi_{x}}$ . For any $x\in\mathbb{R}$ , (7) yields the existence of a Gaussian process $\{\mathcal{G}^{\Phi}(x);\ x\in\mathbb{R}\}$ such that the following convergence holds:

(9) \begin{align} \frac{\mathcal Z^{\Phi}(x+n)-\mathcal{F}^{\Phi}(n+x)}{n^{\ell+\frac12}\rho^{n/2}\sqrt{W}}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} \mathcal{G}^{\Phi}(x), \quad \text{as } n\to\infty.\end{align}

The Cramér–Wold device implies that, in fact, the convergence holds for finite-dimensional distributions and the limiting process $\mathcal{G}^{\Phi}$ is jointly Gaussian with $\mathcal{G}^{\Phi}(x)\overset{\textrm{d}}{=} \rho^{\lfloor{x}\rfloor/2}\sigma_\ell(\Phi_{\{x\}}) \mathcal N(0,1)$ or $\mathcal{G}^{\Phi}(x)\overset{\textrm{d}}{=} \rho^{\lfloor{x}\rfloor/2}\sigma(\Phi_{\{x\}}) \mathcal N(0,1)$ depending on the value of the constants $\sigma_\ell$ or $\sigma$ , respectively. Furthermore, we write $\mathcal{Z}^{\mathsf{t}}, \mathcal{F}^{\mathsf{t}},\mathcal{G}^{\mathsf{t}}$ (respectively $\mathcal{Z}^{j},\mathcal{F}^{j},\mathcal{G}^{j}$ ) for $\mathcal{Z}^{\Phi}, \mathcal{F}^{\Phi},\mathcal{G}^{\Phi}$ if $\Phi=\Phi^{\mathsf{t}}$ (respectively $\Phi=\Phi^j$ ). Since, with probability one, all the random variables $(U_u)_{u\in \mathcal{U}_{\infty}}$ are different, the process $(\mathcal{Z}^{\mathsf{t}}(x))_{x\geq 0}$ at its jump point increases by 1. Therefore, the following stopping times are well defined: for $k\in\mathbb{N}$ , define $\tau_k$ as

(10) \begin{equation}\tau_k=\inf\big\{x\geq 0\;:\; \mathcal{Z}^{\mathsf{t}}(x)=k\big\}.\end{equation}

Remark 1. With the stopping times $(\tau_k)_{k\in\mathbb{N}}$ just introduced, we have

\begin{align*}B_j(k)=\mathcal{Z}^{j}(\tau_k),\end{align*}

and this is exactly the number of balls of type j added to the two urns after k draws, for which we seek first- and second-order asymptotics. Our strategy is as follows: first we prove functional limit theorems for the processes $\{\mathcal{Z}^{\mathsf{t}}(x);\ x\in \mathbb{R}\}$ and $\{\mathcal{Z}^{j}(x);\ x\in\mathbb{R}\}$ , and then we conclude the corresponding limit theorems for $B_j(k)$ , for $j\in[J]$ . We start with the description of the leading term in the asymptotics of $\mathcal{Z}^{\Phi}(x)$ , for any characteristic $\Phi\in \mathcal{C}$ .

Periodic functions. For any $\lambda\in \mathbb{C}$ , we introduce the function

(11) \begin{equation}l_\lambda\;:\;[0,\infty)\to\mathbb{R}\quad \text{defined as}\quad l_\lambda(x)= (1+(\lambda -1)\{x\})\lambda^{-\{x\}},\end{equation}

where $\lambda^t=e^{t\log \lambda}$ and $z\mapsto \log z$ is the principal branch of the logarithm. The function $l_\lambda$ is continuous and 1-periodic, and it satisfies

\begin{align*}\lambda^xl_{\lambda}(x)=\lambda^{\lfloor x\rfloor}(1+(\lambda-1)\{x\}).\end{align*}

Moreover, the mapping $x\mapsto \lambda^xl_{\lambda}(x)$ equals $\lambda^x$ for integer x and is linear in between.

Proposition 2. Assume (GW1), (GW2), and (GW4) hold, and for any $j\in[J]$ let $\Phi=a\Phi^{\mathsf{t}}+b\Phi^j$ , with $a,b\in\mathbb{R}$ . Then it holds that

(12) \begin{align} \lim_{x\to\infty}\frac{\mathcal{Z}^\Phi(x)}{\rho^{x}l_{\rho}(x)}=\lim_{x\to\infty}\frac{\mathcal{Z}^\Phi(x)}{\rho^{\lfloor x\rfloor}(1+(\rho-1)\{x\})}=(a+b\rho\textrm{u}_j)\frac{1}{\rho-1}W,\quad \mathbb{P}^\mathcal{S}\textit{-almost surely}. \end{align}

Proof. Because of the linearity of CMJ processes, for $x\in \mathbb{R}$ it holds that

\begin{align*}\mathcal{Z}^{\Phi}(x)=a\mathcal{Z}^{\mathsf{t}}(x)+b\mathcal{Z}^{j} (x)\quad \text{ for any } a,b\in\mathbb{R}\ \text{and }j\in[J],\end{align*}

so it suffices to prove the $\mathbb{P}^\mathcal{S}$ -almost sure convergence for $\mathcal{Z}^{\mathsf{t}}(x)$ and $\mathcal{Z}^{j}(x)$ separately, as $x\to\infty$ .

Case 1: $x\in[0,1)$ . For $n\in\mathbb{N}$ and $x\in[0,1)$ , since $\mathcal{Z}^{\mathsf{t}}(x+n)=Z_n^{\Phi^{\mathsf{t}}_x}$ , in view of Proposition 2.1 we get

(13) \begin{equation}\frac{\mathcal{Z}^{\mathsf{t}}(x+n)}{\rho^n(1+(\rho-1)x)}\to\frac{1}{\rho-1}W,\quad \mathbb{P}^\mathcal{S}\text{-almost surely as } n\to\infty.\end{equation}

Case 2: $x\in[0,\infty)$ . This case can be reduced to the previous one, where $x\in[0,1)$ , as follows. In view of Equation (13), for any $x\in [0,\infty)$ , with $m=n+\lfloor x \rfloor$ we obtain

\begin{align*}\frac{\mathcal{Z}^{\mathsf{t}}(x+n)}{\rho^{x+n}l_\rho(x)}=\frac{\mathcal{Z}^{\mathsf{t}}(\lfloor x\rfloor+n+\{x\})}{\rho^{\lfloor x\rfloor+n}(1+(\rho-1)\{x\})}=\frac{\mathcal{Z}^{\mathsf{t}}(\{x\}+m)}{\rho^{m}(1+(\rho-1)\{x\})}\xrightarrow{\scriptscriptstyle{a.s.}} \frac{1}{\rho-1}W,\textit{ as } m\to \infty.\end{align*}

In the last equation above, we used the fact that

\begin{align*}\rho^{x+n}l_\rho(x)=\rho^{\lfloor x\rfloor+n}(1+(\rho-1)\{x\})=\rho^{x+n}l_\rho(x+n).\end{align*}

We still have to prove that the above $\mathbb{P}^\mathcal{S}$ -almost sure convergence holds for $x\to\infty$ , that is, that

\begin{align*}\lim_{x\to\infty}\frac{\mathcal{Z}^{\mathsf{t}}(x)}{\rho^{x}l_{\rho}(x)}=\frac{1}{\rho-1}W,\quad \mathbb{P}^\mathcal{S}\text{-almost surely}.\end{align*}

Indeed, from any sequence tending to infinity we may choose a subsequence $(x_n)$ such that $\{x_n\}$ converges to some $x_0\in\mathbb{R}$ . Then for any $\delta>0$ and large n, in view of

\begin{align*}\mathcal{Z}^{\mathsf{t}}(\lfloor x_n\rfloor+ x_0-\delta)=Z_{\lfloor\lfloor x_n\rfloor+ x_0-\delta\rfloor}^{\Phi^{\mathsf{t}}_{\{\lfloor x_n\rfloor+x_0-\delta\}}}=Z_{\lfloor\lfloor x_n\rfloor+ x_0-\delta\rfloor}^{\Phi^{\mathsf{t}}_{\{x_0-\delta\}}},\end{align*}

we have

\begin{align*}&\frac{\mathcal{Z}^{\mathsf{t}}(\lfloor x_n\rfloor+ x_0-\delta)}{\rho^{\lfloor x_n\rfloor+ x_0-\delta}l_\rho(x_0-\delta)}\cdot\rho^{x_0-\{x_n\}-\delta} \cdot\frac{l_{\rho}(x_0-\delta)}{l_{\rho}(x_n)}\le\frac{\mathcal{Z}^{\mathsf{t}}(x_n)}{\rho^{x_n}l_{\rho}(x_n)}\\[5pt] &\le\frac{\mathcal{Z}^{\mathsf{t}}(\lfloor x_n\rfloor+ x_0+\delta)}{\rho^{\lfloor x_n\rfloor+ x_0+\delta}l_{\rho}(x_0+\delta)}\rho^{x_0-\{x_n\}+\delta}\cdot\frac{l_{\rho}(x_0+\delta)}{l_{\rho}(x_n)}.\end{align*}

Taking the limit first as n goes to infinity and then as $\delta$ goes to 0, we get the desired convergence, since $x\mapsto l_{\rho}(x)$ is uniformly continuous. The same argument can be used to show that for any $j\in[J]$ we have

\begin{align*}\lim_{x\to\infty}\frac{\mathcal{Z}^{j}(x)}{\rho^{x}l_\rho(x)}=\frac{1}{\rho-1}W\rho\textrm{u}_j, \quad \mathbb{P}^\mathcal{S}\text{-almost surely},\end{align*}

and this proves the claim.

An immediate consequence of Proposition 2 is the following corollary.

Corollary 1. Under the assumptions of Proposition 2, for $\Phi={\rho}\textrm u_j\Phi^{\mathsf{t}}-\Phi^j$ , we have

\begin{align*} \lim_{x\to\infty}\frac{\mathcal{Z}^\Phi(x)}{\rho^{x}l_\rho(x)}=\lim_{x\to\infty}\frac{\mathcal{Z}^\Phi(x)}{\rho^{\lfloor x\rfloor}(1+(\rho-1)\{x\})}=0,\quad \mathbb{P}^\mathcal{S}\textit{-almost surely}. \end{align*}

Also, the strong law of large numbers for $(B_j(k))_{k\in\mathbb{N}}$ follows immediately from Proposition 2.

Proof of Theorem 1. Since $\tau_k$ goes to infinity as k does, we have

\begin{align*}\lim_{k\to\infty}\frac{B_j(k)}{k}&=\lim_{k\to\infty}\frac{\mathcal{Z}^{j}(\tau_k)}{\mathcal{Z}^{\mathsf{t}}(\tau_k)}=\lim_{k\to\infty}\frac{\mathcal{Z}^{j}(\tau_k)}{\rho^{\lfloor \tau_k\rfloor}(1+(\rho-1)\{\tau_k\})}\cdot \frac{\rho^{\lfloor \tau_k\rfloor}(1+(\rho-1)\{\tau_k\})}{\mathcal{Z}^{\mathsf{t}}(\tau_k)}\\[5pt] &=\frac{1}{\rho-1}W\rho\textrm{u}_j\cdot \frac{1}{\frac{1}{\rho-1}W}=\rho\textrm{u}_j,\quad \mathbb{P}^\mathcal{S}\text{-almost surely},\end{align*}

and this finishes the proof.

4. Proof of the main result

The proof is completed in several steps:

• • In Lemma 1 we investigate compositions of the fluctuations $\mathcal{F}^{\mathsf{t}}$ and $\mathcal{F}^{j}$ .

• • In Theorem 3 we prove weak convergence of the processes $\mathcal{X}^{\mathsf{t}}=\mathcal{Z}^{\mathsf{t}}-\mathcal{F}^{\mathsf{t}}$ and $\mathcal{X}^j=\mathcal{Z}^{j}-\mathcal{F}^{j}$ (rescaled appropriately) to Gaussian processes $\mathcal{G}^{\mathsf{t}}$ and $\mathcal{G}^{j}$ respectively, for any $j\in[J]$ .

• • Continuity and strict positivity of the variances of the limiting processes $\mathcal{G}^{\mathsf{t}}$ and $\mathcal{G}^{j}$ are analyzed in Proposition 4 and in Lemma 2.

• • Localization of the stopping times $\tau_n$ is done in Proposition 5.

• • Finally, the limit theorems for $B_j(n)$ are given in Proposition 6 and in Theorem 4.

4.1. Leading asymptotic terms

We start by describing the leading terms in the asymptotics of $\mathcal{Z}^{\mathsf{t}}$ and of $\mathcal{Z}^{j}$ for any $j\in[J]$ . We recall first that for a characteristic $\Phi\in\mathcal{C}$ , the leading term in the asymptotics of $\mathcal{Z}^{\Phi}$ is given by $\mathcal{F}^{\Phi}$ ; for simplicity of notation, for $x\in\mathbb{R}$ we write

\begin{align*}\mathcal{X}^{\Phi}(x)=\mathcal{Z}^{\Phi}(x)-\mathcal{F}^{\Phi}(x).\end{align*}

In particular, for $\Phi=\Phi^{\mathsf{t}}$ and $\Phi=\Phi^j$ respectively, we write

\begin{align*}\mathcal{X}^{\mathsf{t}}(x)=\mathcal{Z}^{\mathsf{t}}(x)-\mathcal{F}^{\mathsf{t}}(x)\quad \text{and} \quad\mathcal{X}^j(x)=\mathcal{Z}^{j}(x)-\mathcal{F}^{j}(x).\end{align*}

Lemma 1. Assume (GW1)–(GW3) hold. Then for sufficiently large arguments the inverse function $\mathcal{F}^{\mathsf{inv}}=(\mathcal{F}^{\mathsf{t}})^{-1} $ is well defined, and for every $j\in[J]$ we have

(14) \begin{align} \lim_{t\to\infty}\sup_{s\ge 1}s^{-1}\Big|\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(t+s)\big)-\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(t)\big)-\rho \textrm{u}_j s\Big| =0,\quad \mathbb{P}^\mathcal{S}\textit{-almost surely}. \end{align}

Proof. For any $k\in\mathbb{N}_0, x\in[0,1)$ , the equality

\begin{align*}\mathbb{E} [\Phi_x^{\mathsf{t}}(k)]=(1-x)\mathbb{E} [\Phi_0^{\mathsf{t}}(k)]+x\mathbb{E}[\Phi_0^{\mathsf{t}}(k+1)]\end{align*}

together with Equation (5) gives us

\begin{align*}F_{n}^{\Phi_{x}^{\mathsf t}}=(1-x)F_{n}^{\Phi_{0}^{\mathsf t}}+xF_{n+1}^{\Phi_{0}^{\mathsf t}};\end{align*}

that is, for any $x\ge0$ , $\mathcal{F}^{\mathsf{t}}(x)$ is a linear interpolation between $\mathcal{F}^{\mathsf{t}}(\lfloor x\rfloor)$ and $\mathcal{F}^{\mathsf{t}}(\lfloor x\rfloor+1)$ . On the other hand, as, in view of (4) and (5), for $n\in\mathbb{N}$ we have

\begin{align*}\mathcal{F}^{\mathsf{t}}(n)=\rho^n\textrm{x}_1(\Phi_{0}^{\mathsf t}) A_1^n\pi_\rho W^{(1)}=\tfrac{\rho^{n}}{\rho-1}W+o(\rho^n),\end{align*}

we conclude that the following holds:

\begin{align*} &\mathcal{F}^{\mathsf{t}}(x)=\tfrac{\rho^{x}}{\rho-1}l_\rho(x)W+o(\rho^x)\text{ and }\\[5pt] &\big(\mathcal{F}^{\mathsf{t}}\big)'(x)=\rho^{\lfloor x\rfloor}W+o(\rho^x)\text{ for }x\notin\mathbb{Z}. \end{align*}

By the same argument, for $j\in[J]$ we obtain

\begin{align*} &\mathcal{F}^{j}(x)=\tfrac{\rho^{x}}{\rho-1}l_\rho(x)\rho\textrm{u}_jW+o(\rho^x)\text{ and }\\[5pt] &\big(\mathcal{F}^{j}\big)'(x)=\rho^{\lfloor x\rfloor}\rho\textrm{u}_jW+o(\rho^x)\text{ for }x\notin\mathbb{Z}. \end{align*}

In particular, $\mathcal{F}^{\mathsf{t}}$ is eventually increasing $\mathbb{P}^\mathcal{S}$ -almost surely and thus the inverse function $(\mathcal{F}^{\mathsf{t}})^{-1}$ is well defined for large arguments. Furthermore, if $\mathcal{F}^{\mathsf{inv}}(t)\notin\mathbb{N}$ and t is large enough then we have

\begin{align*}\big(\mathcal{F}^{j}\circ \mathcal{F}^{\mathsf{inv}}\big)'(t)=(\mathcal{F}^{j})'\big(\mathcal{F}^{\mathsf{inv}}(t)\big) \cdot (\mathcal{F}^{\mathsf{inv}}(t))'=\frac{(\mathcal{F}^{j})'\big(\mathcal{F}^{\mathsf{inv}}(t)\big)}{(\mathcal{F}^{\mathsf{t}})'\big(\mathcal{F}^{\mathsf{inv}}(t)\big)}\to\rho\textrm{u}_j,\quad \text{as }t\to\infty,\end{align*}

since $\mathcal{F}^{\mathsf{inv}}(t)$ diverges to infinity. Finally, since for large t it holds that

\begin{align*}\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(t+s)\big)-\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(t)\big)=\int_t^{t+s}\big(\mathcal{F}^{j}\circ \mathcal{F}^{\mathsf{inv}}\big)'(u)du,\end{align*}

we obtain (14).

4.2. Limit theorems for $\mathcal{X}^{\mathsf{t}}$ and $\mathcal{X}^{j}$

To prove weak convergence of the processes $\Big\{\frac{1}{n^{\ell+\frac12}\rho^{n/2}\sqrt W}\mathcal{X}^{j}(n+x); \ x\in \mathbb{R}\Big\}$ , we follow the well-known technique: we first prove weak convergence of the finite-dimensional distributions, then prove tightness. According to [8, Theorem 3.5], the finite-dimensional distributions of the aforementioned processes converge jointly; see also Equation (9) and the discussion thereafter.

Theorem 3. Suppose that (GW1)–(GW3) hold and L satisfies $\mathbb{E}\left[\|L\|^{2+\delta}\right]<\infty$ for some $\delta\in (0,1)$ . Then for every $j\in [J]$ , the family of distributions

\begin{align*}\bigg\{\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{j}(n+x); \ x\in \mathbb{R}\bigg\}\end{align*}

with respect to $\mathbb{P}$ is tight in the Skorokhod space $\mathcal D(\mathbb{R})$ endowed with the standard $\textbf{J}_1$ topology.

Proof. First let us observe that for any $k\in\mathbb{Z},\ m\in\mathbb{N}$ , the concatenation is a continuous mapping from $\mathcal{D}([k,k+1))\times\cdots\times\mathcal D([k+m-1,k+m))$ to $\mathcal D([k,k+m))$ . Consequently, it suffices to prove tightness in the space $\mathcal D([0,1))$ . For $x\in[0,1)$ , $j\in[J]$ , and $n\in\mathbb{N}$ we set

\begin{align*}\mathcal{Y}^{j}(n+x)=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{j}(n+x),\end{align*}

where $\ell$ may be $-\frac12$ in Case (i) of [8, Theorem 3.5], and the characteristic $\Phi^j=(\Phi^j_x)_{x\in[0,1)}$ is defined as in (3). In view of [3, Theorem 13.5], it suffices to show that for any $ 0\leq x\le y\le z<1$ , $\lambda>0$ , and n large enough, it holds that

\begin{align*}\mathbb{P}\Big(\big|\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)\big|\wedge\big|\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y)\big|\ge \lambda\Big)\le C\lambda^{-2p}|z-x|^{p}, \end{align*}

where $p:==(2+\delta)/2\in (1,3/2)$ and $C>0$ is some constant. A more restrictive condition is the following inequality:

\begin{align*}\mathbb{E}\Big[\left |\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)\right|^p\cdot\left |\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y))\right|^p\Big]\le C|z-x|^{p}.\end{align*}

By Hölder’s inequality, we have

\begin{align*}\mathbb{E}&\Big[|\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)|^{p}\cdot |\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y)|^{p}\Big]\\[5pt] &\le\mathbb{E}\left[\mathbb{E}\Big[|\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)|^{2}\cdot |\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y)|^{2}\Big|\mathcal{A}_n\Big]^{p/2}\right], \end{align*}

so it remains to show the estimate

\begin{align*}\mathbb{E}\Big[|\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)|^{2}\cdot |\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y)|^{2}\Big|\mathcal{A}_n\Big]\leq H_n|z-x|^2,\end{align*}

for some random variables $H_n$ with bounded $p/2$ moment. Recalling that for $0\leq x<1$ we have

\begin{align*}\mathcal{Z}^{j}(n+x)=\sum_{k=0}^{n-1}Z_k^j+\sum_{u\in\mathbb{T}; |u|=n+1; \textrm t(u)=j}\textbf{1}_{\{U_u\leq x\}},\end{align*}

we deduce that for $0\leq x\leq y<1$ ,

\begin{align*}\mathcal{Z}^{j}(n+y)-\mathcal{Z}^{j}(n+x)=\sum_{u\in \mathbb{T};|u|=n+1,\textrm t(u)=j}\textbf{1}_{\{x< U_u\leq y\}}.\end{align*}

We also have

\begin{align*}\mathcal{F}^{j}(n+y)-\mathcal{F}^{j}(n+x)&=\left(\textrm{x}_1(\Phi^j_y)-\textrm{x}_1(\Phi^j_x)\right)A^n_1 W^{(1)}+\left(\textrm{x}_2(\Phi^j_y)-\textrm{x}_2(\Phi^j_x)\right)A^n_2Z_0\\[5pt] &=(y-x)A\textrm{e}_j^\top\pi^{(1)}A^n_1 W^{(1)}+(y-x)A\textrm{e}_j^\top\pi^{(2)}A^n_2Z_0\\[5pt] &=(y-x)A\textrm{e}_j^\top\big(\pi^{(1)}A^n_1 W^{(1)}+\pi^{(2)}A^n_2Z_0\big)\\[5pt] &=(y-x)F_n^{\Psi}=(y-x)(Z^j_{n+1}-(Z_n^{\Psi}-F_n^{\Psi})),\end{align*}

where $\Psi\;:\;\mathbb{N}_0\to\mathbb{R}^{1\times J}$ is the characteristic given by $\Psi(k)=\textbf{1}_{\{k=0\}}\textrm{e}_j^{\top}L=\textbf{1}_{\{k=0\}}\langle\textrm{e}_j,L\rangle$ . Hence $Z_n^{\Psi}$ counts the number of individuals of type j in the $(n+1)$ th generation. We then obtain

\begin{align*}\mathcal{Y}^{j}&(n+y)- \mathcal{Y}^{j}(n+x)=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\left(\mathcal{X}^{j}(n+y)-\mathcal{X}^{j}(n+x)\right)\\[5pt] &=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\left[(\mathcal{Z}^{j}(n+y)-\mathcal{Z}^{j}(n+x))-(\mathcal{F}^{j}(n+y)-\mathcal{F}^{j}(n+x))\right]\\[5pt] &=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\Big[\sum_{u\in\mathbb{T}; |u|=n+1; \textrm t(u)=j}\textbf{1}_{\{x< U_u\leq y\}}-(y-x)Z^j_{n+1}+(y-x)(Z_n^{\Psi}-F_n^{\Psi})\Big]\\[5pt] &=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\Big[\sum_{u\in\mathbb{T}; |u|=n+1; \textrm t(u)=j}\left(\textbf{1}_{\{x< U_u\leq y\}}-(y-x)\right)\cdot 1+(y-x)(Z_n^{\Psi}-F_n^{\Psi})\Big].\end{align*}

Since $Z_{n+1}^j$ and $Z_n^{\Psi}-F_n^{\Psi}$ are $\mathcal{A}_n$ -measurable, by applying Lemma 5 to the intervals $I=(x,y)$ , $J=(y,z)$ , with $a_i=1$ , $A=(y-x)(Z_n^{\Psi}-F_n^{\Psi})$ , and $B=(z-y)(Z_n^{\Psi}-F_n^{\Psi})$ , we finally obtain

\begin{align*}\mathbb{E} & \Big[|\mathcal{Y}^{j}(n+y)-\mathcal{Y}^{j}(n+x)|^{2}\cdot |\mathcal{Y}^{j}(n+z)-\mathcal{Y}^{j}(n+y)|^{2}|\mathcal{A}_n\Big]\\[5pt] &\le C \Big(\frac{1}{n^{2\ell+1}\rho^{n}}\Big)^2\big((y-x)(z-y)\big(Z_{n+1}^j\big)^2 +A^4+B^4\big)\\[5pt] & \leq C (z-x)^2 \Big(\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\Big)^4\left[(Z_n^{\Psi}-F_n^{\Psi})^4+(Z_{n+1}^j)^2\right]=:C|z-x|^2H_n,\end{align*}

for some absolute constant C. We have $Z_{n+1}^j=Z_n^{\Psi}$ , so Theorem 5(i) implies that $\mathbb{E}[(Z_{n+1}^j)^p]=O(\rho^{pn})$ . On the other hand, Corollary 4 yields that $\mathbb{E}\big[(Z_n^{\Psi}-F_n^{\Psi})^{2p}\big] =O(n^{(2\ell+1)p}\rho^{pn})$ . As a consequence, the random variables $H_n$ have bounded $p/2$ moments, and in turn the process $\Big\{\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{j}(n+x); \ x\in \mathbb{R}\Big\}$ is tight in $\mathcal D(\mathbb{R})$ .

As a consequence of the previous result we obtain the following.

Corollary 2. Suppose that the assumptions (GW1)–(GW3) hold and that the matrix L satisfies $\mathbb{E}\left[\|L\|^{2+\delta}\right]<\infty$ for some $\delta\in (0,1)$ . Then the family of distributions

\begin{align*}\Big\{\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{\mathsf{t}}(n+x); \ x\in \mathbb{R}\Big\}\end{align*}

is tight in the Skorokhod space $\mathcal D(\mathbb{R})$ endowed with the standard $\textbf{J}_1$ topology.

Proof. In view of the two equalities $\mathcal{Z}^{\mathsf{t}}(n+x)-1=\sum_{j=1}^J \mathcal{Z}^{j}(n-1+x)$ and $\mathcal{F}^{\mathsf{t}}(n+x)=\sum_{j=1}^J \mathcal{F}^{j}(n-1+x)$ , together with

\begin{align*}\mathcal{Y}^{\mathsf{t}}(n+x)=\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{\mathsf{t}}(n+x)=\sum_{j=1}^J\frac{1}{n^{\ell+\frac12}\rho^{n/2}}\mathcal{X}^{j}(n-1+x),\end{align*}

we see that $\mathcal{Y}^{\mathsf{t}}(n+x)$ can be written as a finite sum of tight processes, so it is tight as well.

The convergence of the finite-dimensional distributions together with the tightness gives the weak convergence.

Proposition 3. Suppose that the assumptions (GW1)–(GW3) hold and that the matrix L satisfies $\mathbb{E}\left[\|L\|^{2+\delta}\right]<\infty$ for some $\delta\in (0,1)$ . Then we have the following weak convergence of sequences of processes in the Skorokhod space $\mathcal D(\mathbb{R})$ endowed with the standard $\textbf{J}_1$ topology: for every $j\in[J]$ it holds that

\begin{align*}\bigg\{\frac{1}{n^{\ell+\frac12}\rho^{n/2}\sqrt W}(\mathcal{X}^{\mathsf{t}}(n+x),\mathcal{X}^{j}(n+x)); \ x\in \mathbb{R}\bigg\}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} \{(\mathcal{G}^{\mathsf{t}}(x),\mathcal{G}^{j}(x));\ x\in\mathbb{R}\}.\end{align*}

4.3. Properties of the limiting processes $\mathcal{G}^{\mathsf{t}}$ and $\mathcal{G}^{j}$

Notice that for $\Phi_0\in\mathcal{C}$ , we have

\begin{align*}\Phi_0(1)=a \Phi_0^{\mathsf{t}}(1)+b\Phi_0^j(1)=a\textrm{1}+b\textrm{e}_j^\top L \quad \text{and} \quad \mathbb{E}[\Phi_0(1)]=a\textrm{1}+b\textrm{e}_j^\top A,\end{align*}

where $a,b\in \mathbb{R}$ and $j\in[J]$ . On the other hand, taking $a=-\rho\textrm{u}_j$ and $b=1$ , we recover

\begin{align*}\textrm{w}_j=\textrm{e}_j^{\top}A-\rho\textrm{u}_j\textrm{1}=\mathbb{E}[\Phi_0^j(1)-\rho\textrm{u}_j\Phi_0^{\mathsf{t}}(1)],\end{align*}

as defined in (1), whose ith entry is given by $\textrm{w}_{ji}=\mathbb{E}[L^{(i,j)}-\rho\textrm{u}_j]=a_{ji}-\rho\textrm{u}_j$ .

Proposition 4. For any $\Phi\in\mathcal C$ and $j\in[J]$ , assume that $\textrm{w}_j\ne \textrm{0}$ and that

(15) \begin{align} \sum_{\lambda\in\sigma_A} \sum_{\ell=0}^{J-1}\|\textrm{Var}(\textrm{w}_jN^\ell\pi_{\lambda}L)\|>0. \end{align}

1. (i) If $\sum_{\lambda\in\sigma_A^{2}}\sum_{\ell=0}^{J-1}\|\textrm{Var}(\textrm{w}_jN^\ell\pi_{\lambda}L)\|>0$ , then for any $x\in[0,1)$ it holds that

   (16) \begin{align}\max\{0\leq \ell\leq J-1\;:\;\sigma_\ell^2(\Phi_x)>0\}=\max\Big\{\ell\ge 0\;:\;\sum_{\lambda\in\sigma^{2}_A}\|\textrm{Var}(\textrm{w}_jN^\ell\pi_{\lambda}L)\|>0\Big\}.\end{align}
   In particular, the largest $\ell$ such that $\sigma_\ell^2(\Phi_x)>0$ does not depend on x.

2. (ii) Otherwise, for any $x\in[0,1)$ and $0\leq \ell\leq J-1$ , it holds that

   \begin{align*}\sigma_\ell^2(\Phi_x)=0 \quad \text{ and } \quad \sigma^2(\Phi_x)>0.\end{align*}

Proof. (i) For any $x\in[0,1)$ , the vector $\textrm{x}_2(\Phi_x)$ is given by

\begin{align*} \textrm{x}_2(\Phi_x)=\sum_{k=0}^{\infty}\mathbb{E}[\Phi_x(k)]\pi^{(2)}A_2^{-k}=x\textrm{w}_j\pi^{(2)}+\textrm{w}_j\pi^{(2)}\sum_{k=1}^{\infty}A_2^{-k}. \end{align*}

If k is at least the right-hand side of Equation (16), then for any $\lambda\in\sigma^{2}_A$ we have $\textrm{w}_jN_\lambda^k(L-A)= 0$ almost surely. Since $A_2$ is invertible, we have

\begin{align*}\sum_{j\geq 1}A_2^{-j}=A_2^{-1}\sum_{j\geq 0}A_2^{-j}=\sum_{j\geq 0}A_2^{-j}-I,\end{align*}

so from the last two matrix equations we get $\sum_{j\geq 0}A_2^{-j}=A_2\sum_{j\geq 0}A_2^{-j}-A_2$ , which in turn implies $(A_2-I)\sum_{j\geq 0}A_2^{-j}=A_2$ . Multiplying this equation by $(A_2-I)^{-1}$ from the right and by $A_2^{-1}$ from the left, we obtain $\sum_{j\geq 1}A_2^{-j}=(A_2-I)^{-1}$ ; hence it holds that $(A_2-I)\sum_{j\geq 1}A_2^{-j}\pi_{\lambda}=\pi_{\lambda}$ , and finally it follows that $\sum_{j\geq 1}A_2^{-j}\pi_{\lambda}=((\lambda-1)I+N_{\lambda})^{-1}$ . In view of the definition (6) of $\sigma_\ell(\Phi)$ , it is enough to understand the term $\textrm{x} _2(\Phi) \pi_\lambda (A-\lambda I)^{\ell}(L-A)$ . It holds that

\begin{align*} \textrm{x} _2(\Phi_x) \pi_{\lambda} (A-\lambda I)^{k}&(L-A)=\Big(x\textrm{w}_j\pi_{\lambda}+\textrm{w}_j\sum_{j=1}^{\infty}A_2^{-j}\pi_{\lambda}\Big) (A-\lambda I)^{k}\pi_{\lambda}(L-A)\\[5pt] &=\big(x\textrm{w}_j\pi_{\lambda}+\textrm{w}_j((\lambda-1)I +N_{\lambda})^{-1}\big) N_{\lambda}^{k}(L-A)\\[5pt] &=\bigg(x\textrm{w}_j\pi_{\lambda}+\textrm{w}_j\sum_{j=0}^{d_{\lambda}-1}\left(\begin{array}{c}{-1}\\[2pt] {j}\end{array}\right)(\lambda-1)^{-1-j}N_{\lambda}^j\bigg) N_{\lambda}^{k}\pi_{\lambda}(L-A)= 0, \end{align*}

almost surely, and hence $\sigma_{\ell}^2(\Phi_x)=0$ . On the other hand, if $0\leq \ell\leq J-1$ is the maximal number such that $\mathbb{P}(\textrm{w}_jN_\lambda^\ell(L-A)\neq 0)>0$ for some $\lambda\in\sigma^{2}_A$ , then for such $\ell$ and $\lambda$ , similar calculations give

\begin{align*} \textrm{x} _2(\Phi_x) \pi_\lambda (A-\lambda I)^{\ell}(L-A) &=\bigg(x\textrm{w}_j\pi_\lambda+\textrm{w}_j\sum_{j=0}^{d_\lambda-1}\left(\begin{array}{c}{-1}\\[2pt] {j}\end{array}\right)(\lambda-1)^{-1-j}N_\lambda^j\bigg) N_\lambda^{\ell}\pi_\lambda(L-A)\\[5pt] &=\textrm{w}_j\big(x+(\lambda-1)^{-1}\big) N_\lambda^{\ell}\pi_\lambda(L-A)\neq 0\\[5pt] \end{align*}

with positive probability, since $x+(\lambda-1)^{-1}\neq0$ because $\lambda\notin\mathbb{R}$ . This implies that $\sigma_\ell^2(\Phi_x)>0$ , and this proves (i).

(ii) Assume that $\textrm{w}_jN^\ell_{\lambda}(L-A)=0$ almost surely for any $\lambda\in\sigma_A^{2}$ and any $\ell\ge 0$ . Let $\textrm{t}_j=\Phi_{0}(1)=a\textrm{1}+b \textrm{e}_j^\top L\in \mathbb{R}^{1\times J}$ be the random row vector whose expectation is $\mathbb{E}[\mathsf{t}_j]=\textrm{w}_j\neq \textrm{0}$ by assumption, and whose ith entry is denoted by $\textrm{t}_{ji}$ . Note that for $x\in(0,1)$ , in view of (8), we have

\begin{align*} \sigma^2(\Phi_x)&=\sum_{k=0}^{\infty}\rho^{-k}\textrm{Var}\left[\Phi_x(k)+\Psi^{\Phi_x}(k)\right]\textrm{u} \ge\textrm{Var}\left[\Phi_x(0)+\Psi^{\Phi_x}(0)\right]\textrm{u}\\[5pt] &\ge \mathbb{E}\big[\textrm{Var}\big[\textrm{t}_j\textbf{1}_{\{U\le x\}}+\Psi^{\Phi_x}(0)\big|L\big]\textrm{u}\big]+\textrm{Var}\big[\mathbb{E}\big[\textrm{t}_j\textbf{1}_{\{U\le x\}}+\Psi^{\Phi_x}(0)\big|L\big]\textrm{u}\big]\\[5pt] &\ge \mathbb{E}\big[\textrm{Var}\big[\textrm{t}_j\textbf{1}_{\{U\le x\}}\big|L\big]\textrm{u}\big]=x(1-x)\mathbb{E}\Big[\sum_{i=1}^J \textrm{t}^2_{ji} \textrm u_i\Big] \ge x(1-x)\sum_{i=1}^J \textrm{w}_{ji}^2 \textrm u_i>0, \end{align*}

since by assumption $\textrm{w}_j\neq0$ . Now we prove that $\sigma^2(\Phi_0)$ does not vanish. We have $\Phi_0(k)=\textrm{w}_j\cdot \textbf{1}_{\{k\geq 1\}}$ , so $\Phi_0\;:\;\mathbb{N}_0\mapsto \mathbb{R}^{1\times J}$ is completely deterministic. Assume that $\sigma^2(\Phi_0)=0$ . Then for any $k\in \mathbb{N}_0$ it holds that $\textrm{Var}\left[\Psi^{\Phi_0}(k)\right]\textrm{u}=0$ , which in turn implies, as $\textrm{u}_j>0$ for $j\in[J]$ , that for any $k\in \mathbb{N}_0$ and $j\in[J]$ we have $\textrm{Var}\left[\Psi^{\Phi_0}(k)\textrm{e}_j\right]=0$ . The latter is equivalent to

(17) \begin{align} \sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA^\ell\mathsf{P}(k,\ell)(L-A)\textrm{e}_j=0\quad\text{almost surely}. \end{align}

We set $A_\lambda = \pi_\lambda A+\lambda(I-\pi_\lambda)$ and observe that for any $n\in\mathbb{Z}$ and $m\in \mathbb{N}$ we have

\begin{align*} A_{\lambda}^n\pi_{\lambda}=(\lambda I+N_\lambda)^n\pi_{\lambda}=\lambda^{n}\sum_{i=0}^{d_{\lambda}}\lambda^{-i}\left(\begin{array}{c}{n}\\[2pt] {i}\end{array}\right)N_{\lambda}^i\pi_{\lambda}\end{align*}

and

\begin{align*} \sum_{0\le \ell\le m}A_{1}^\ell\pi_{1}&=\sum_{0\le \ell\le m}(I+N_1)^ \ell\pi_{1}=\sum_{0\le \ell\le m}\sum_{i=0}^\ell\left(\begin{array}{c}{\ell}\\[2pt] {i}\end{array}\right)N_1^i\pi_{1} =\sum_{i=0}^{ d_1}N_1^i\sum_{l=i}^{m}\left(\begin{array}{c}{l}\\[2pt] {i}\end{array}\right)\pi_{1}, \end{align*}

where in the second equation we have used $\lambda=1$ (if $\lambda$ would be in the spectrum of A). Suppose now that for some vector $\textrm{z}\in\mathbb{R}^J$ we have

\begin{align*} \textrm{w}_jN^i_\lambda\pi_\lambda \textrm{z}=0\quad\text{for any }\lambda\in\sigma^{2}_A \text{ and }\ i=0,\ldots,d_\lambda-1, \end{align*}

and for any $k\in\mathbb{N}_0$ it holds that

(18) \begin{align} \sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA^\ell\mathsf{P}(k,\ell)\textrm{z}=0. \end{align}

Note that the left-hand side of the previous equation can be rewritten as

\begin{align*} &\sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA^\ell\mathsf{P}(k,\ell)\textrm{z}= \sum_{\lambda\in\sigma_A}\sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\mathsf{P}(k,\ell)\textrm{z}\\[5pt] &=\sum_{\lambda\in\sigma_A^{1}\cup\sigma_A^{3}}\sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\mathsf{P}(k,\ell)\textrm{z}\\[5pt] &=-\sum_{\lambda\in\sigma^{1}_A}\sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\textbf{1}_{\{\ell<0\}}\textrm{z} +\sum_{\lambda\in\sigma^{3}_A}\sum_{\ell\in \mathbb{Z}}\textbf{1}_{\{k-\ell-1\ge 1\}}\textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\textbf{1}_{\{\ell\ge 0\}}\textrm{z}. \end{align*}

Setting $-n=k-2\ge -2$ , we get

\begin{align*} 0&=\sum_{\lambda\in\sigma^{1}_A}\sum_{\ell\le -n}\textrm{w}_jA_{\lambda}^\ell\textbf{1}_{\{\ell<0\}}\pi_{\lambda}\textrm{z} =\sum_{\lambda\in\sigma^{1}_A}\textrm{w}_j(I-A_{\lambda}^{-1})^{-1}A_{\lambda}^{-n}\pi_{\lambda}\textrm{z}\\[5pt] &=\sum_{\lambda\in\sigma^{1}_A}\textrm{w}_j(I-A_{\lambda}^{-1})^{-1}(\lambda\pi_{\lambda}+N_\lambda)^{-n}\pi_{\lambda}\textrm{z}\\[5pt] &=\sum_{\lambda\in\sigma^{1}_A}\lambda^{-n}\sum_{i=0}^{d_{\lambda}}\left(\begin{array}{c}{-n}\\[2pt] {i}\end{array}\right)\lambda^{-i}\Big(\textrm{w}_j(I-A_{\lambda}^{-1})^{-1}N_{\lambda}^i\pi_{\lambda}\textrm{z}\Big),\end{align*}

which in view of Lemma 6 implies that

(19) \begin{align} \textrm{w}_j(I-A_{\lambda}^{-1})^{-1}N^i_\lambda\pi_\lambda \textrm{z}=0&\quad\text{for }\lambda\in\sigma_A^{1} \text{ and }\ i<d_\lambda. \end{align}

Now we can use the decomposition

\begin{align*}(I-A_{\lambda}^{-1})^{-1}\pi_\lambda=\sum_{i=0}^{d_\lambda-1}c_iN^i_\lambda\pi_\lambda,\end{align*}

for some $c_0,\ldots,c_{d_\lambda-1}$ , and since $(I-A_{\lambda}^{-1})^{-1}$ is invertible, we conclude that the matrix $(I-A_{\lambda}^{-1})^{-1}\pi_\lambda$ is not nilpotent and hence $c_0\neq0$ . Now, taking $i=d_{\lambda-1}$ in (19), we infer that $c_0\textrm{w}_jN_\lambda^{d_{\lambda}-1}\textrm{z}=0$ ; that is, we have $\textrm{w}_jN_{\lambda}^{d_{\lambda}-1}\textrm{z}=0$ . Recursively, we see that for $i=d_{\lambda}-1,d_{\lambda}-2,\ldots,0$ Equation (19) implies $\textrm{w}_jN_\lambda^i\textrm{z}=0$ , for any $\lambda\in\sigma_A^{1}$ . Taking this into account and setting $n=k-2>0 $ , from the condition (18) we get

\begin{align*} 0&=\sum_{\lambda\in\sigma^3_A}\sum_{0\le \ell\le n} \textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\textrm{z} =\sum_{\lambda\in\sigma^3_A\setminus\{1\}}\sum_{0\le \ell\le n} \textrm{w}_jA_{\lambda}^\ell\pi_{\lambda}\textrm{z} +\textbf{1}_{\{1\in\sigma^{3}_A\}}\sum_{0\le \ell\le n} \textrm{w}_jA_{1}^\ell\pi_{1}\textrm{z}\\[5pt] &=\sum_{\lambda\in\sigma^3_A\setminus\{1\}} \textrm{w}_j(A_{\lambda}-I)^{-1}(A_{\lambda}^{n+1}-I)\pi_{\lambda}\textrm{z} +\textbf{1}_{\{1\in\sigma^{3}_A\}}\sum_{0\le \ell\le n} \textrm{w}_j(I+N_1)^\ell\pi_{1}\textrm{z}\\[5pt] &=\sum_{\lambda\in\sigma^3_A\setminus\{1\}}\sum_{i=0}^{d_\lambda-1}\textrm{w}_j(A_{\lambda}-I)^{-1}N^i_\lambda\pi_\lambda \textrm{z}\lambda^np_{\lambda,i}(n)+\textbf{1}_{\{1\in\sigma^{3}_A\}} \sum_{i=0}^{d_1-1}\textrm{w}_jN^i_1\pi_1 \textrm{z} p_{1,i+1}(n)+c, \end{align*}

where $p_{\gamma,i}$ is some polynomial of degree i and c does not depend on n. Lemma 6 implies that

\begin{align*} \textrm{w}_j(A_{\lambda}-I)^{-1}N^i_\lambda\pi_\lambda \textrm{z}=0\quad\text{for }\lambda\in\sigma_A^{3}\setminus\{1\} \text{ and }i<d_\lambda, \end{align*}

and also

\begin{align*} \textrm{w}_jN^i_1\pi_1 \textrm{z}=0\quad\text{if }1\in\sigma_A^{3} \text{ and } i<d_1. \end{align*}

The same argument as before gives that $\textrm{w}_jN_\lambda^i\textrm{z}=0,$ for any $\lambda\in\sigma_A^{3}$ and $i<d_\lambda$ . Suppose now that $\sigma^2(\Phi_0)=0$ and also $\sigma_\ell^2(\Phi_0)=0$ for all $0\leq \ell\leq J$ . Then by setting $\textrm{z}=(L-A)\textrm e_j$ with $j\in [J]$ we conclude that for any $\lambda\in \sigma_A$ and $\ell\ge 0$ ,

\begin{align*} \textrm{w}_jN^\ell_\lambda\pi_\lambda L=\textrm{w}_jN^\ell_\lambda\pi_\lambda A\quad \text{almost surely}. \end{align*}

This contradicts the assumption.

Continuity of the limit processes $\boldsymbol{\mathcal{G}}^{\mathsf{t}}$ and $\boldsymbol{\mathcal{G}}^{\textbf{\textit{j}}}$

We recall once again the notation $\mathcal{G}^{\Phi}(x)=\rho^{\lfloor x\rfloor/2}G^{\Phi_{\{x\}}}$ , where $G^{\Phi_{\{x\}}}\overset{\textrm{d}}{=}\sigma_\ell(\Phi_{\{x\}})\mathcal{N}$ or $G^{\Phi_{\{x\}}}\overset{\textrm{d}}{=}\sigma(\Phi_{\{x\}})\mathcal{N}$ , with $\mathcal{N}\;:\!=\;\mathcal{N}(0,1)$ denoting a standard normal variable independent of W, and $\mathcal{G}^{\mathsf{t}}$ (respectively $\mathcal{G}^{\Phi}$ ) denoting $\mathcal{G}^{\Phi}$ if $\Phi=\Phi^{\mathsf{t}}$ (respectively $\Phi=\Phi^i$ ).

Lemma 2. For $j\in[J]$ , let $\mathcal{H}(x)=\rho \textrm{u}_j \mathcal{G}^{\mathsf{t}}(x)-\mathcal{G}^{j}(x)$ . Under the assumptions of Theorem 3, $\mathcal{H}(x)$ is continuous for any $x\in [0,\infty)$ .

Proof. Since $\mathcal{H}$ is a linear combination of the Gaussian processes $\mathcal{G}^{\mathsf{t}}$ and $\mathcal{G}^{j}$ , it is enough to show continuity for the two terms separately. We start with the continuity of $\mathcal{G}^{j}$ . For any $0\le x\le y\le 1$ , since either

\begin{align*}\mathcal{G}^{j}(y)-\mathcal{G}^{j}(x)\overset{\textrm{d}}{=}(\sigma(\Phi^j_{y})-\sigma(\Phi^j_x))\mathcal{N}=\sigma(\Phi^j_{y}-\Phi^j_{x})\mathcal{N}\end{align*}

or

\begin{align*}\mathcal{G}^{j}(y)-\mathcal{G}^{j}(x)\overset{\textrm{d}}{=}(\sigma_\ell(\Phi^j_{y})-\sigma_\ell(\Phi^j_x))\mathcal{N}=\sigma_\ell(\Phi^j_{y}-\Phi^j_{x})\mathcal{N},\end{align*}

depending on whether we are in Case (i) or Case (ii) of Theorem 3.5 in [8], we have to upper-bound $\sigma^2(\Phi^j_{y}-\Phi^j_{x})$ and $\sigma_\ell^2(\Phi^j_{y}-\Phi^j_{x})$ by some power of $|y-x|$ . The definition of $\sigma^2(\Phi)$ applied to the characteristic $\Phi^j_{y}-\Phi^j_{x}$ yields

\begin{align*} &\sigma^2(\Phi^j_{y}-\Phi^j_{x})=\sum_{k<0}\rho^{-k}\textrm{Var}[\textrm{e}_j\pi^{(1)}(y-x)A^{k-1})(L-A)]\textrm{u} \\[5pt] + &\textrm{Var}[\textrm{e}_j\textbf{1}_{\{x< U\le y\}}-\textrm{e}_j\pi^{(1)}(y-x)A^{-1}(L-A)]\textrm{u} +\rho\textrm{Var}[(y-x)e_j\pi^{(3)}(L-A)]\textrm{u} \le C(y-x). \end{align*}

On the other hand, as for $\Phi^j_{y}-\Phi^j_{x}$ it holds that $\textrm{x}_2=\textrm{x}_2(\Phi^j_{y}-\Phi^j_{x})=(y-x)\textrm{e}_j\pi^{(2)}$ , we conclude that

\begin{align*} \sigma_\ell^2(\Phi^j_{y}-\Phi^j_{x})= \frac{\rho^{-\ell}}{(2\ell+1)(\ell!)^2}\sum_{\lambda\in\sigma^2_A}\textrm{Var} \left[\textrm{x} _2 \pi_\lambda (A-\lambda I)^{\ell}L\right]\textrm{u}\le C(y-x)^2. \end{align*}

In particular, in both of the cases (i) and (ii) of [8, Theorem 3.5] we have

\begin{align*}\mathbb{E}[|\mathcal{G}^{j}(y)-\mathcal{G}^{j}(x)|^{2}]\le C(y-x),\end{align*}

and therefore, since $\mathcal{G}^{j}$ is Gaussian, we obtain

\begin{align*}\mathbb{E}[|\mathcal{G}^{j}(y)-\mathcal{G}^{j}(x)|^{4}]=6\mathbb{E}[|\mathcal{G}^{j}(y)-\mathcal{G}^{j}(x)|^{2}]^2\le C|y-x|^2,\end{align*}

which by the Kolmogorov continuity theorem implies that $\mathcal{G}^{j}$ is continuous. The same calculations as for $\mathcal{G}^{j}$ can be carried out to prove that $\mathcal{G}^{\mathsf{t}}$ is continuous, so also $\mathcal{H}(x)=\rho\textrm{u}_j\mathcal{G}^{\mathsf{t}}(x)-\mathcal{G}^{j}(x)$ is continuous.

Localization of the stopping times

This section addresses the localization of the stopping times $(\tau_k)_{k\in\mathbb{N}}$ . On the non-extinction event $\mathcal{S}$ , for any $n\in\mathbb{N}$ , we define the random variable

\begin{align*}T_n=\log_\rho\frac{n(\rho-1)}{W},\end{align*}

and we define the function $h\;:\;\mathbb{R}\to\mathbb{R}$ by

(20) \begin{equation} h(x)= \lfloor x\rfloor+\frac{\rho^{\{x\}}-1}{\rho-1}=x+\frac{\rho^{\{x\}}-1}{\rho-1}-\{x\}. \end{equation}

Note that $h^{-1}$ is uniformly continuous and given by $h^{-1}(x)=\lfloor x\rfloor+\log_\rho\big(1+(\rho-1)\{x\}\big)$ .

Proposition 5. Under the assumptions (GW1)–(GW3), and if for $k\in\mathbb{N}$ we set $t_k=h(T_k)$ , then for $(\tau_k)$ defined as in (10), we have

\begin{align*}\lim_{k\to\infty}(t_k-\tau_k)=\lim_{k\to\infty}(h(T_k)-\tau_k)=0,\quad\mathbb{P}^\mathcal{S}\textit{-almost surely}.\end{align*}

Proof. By Proposition 2, the following $\mathbb{P}^\mathcal{S}$ -almost sure convergence holds:

\begin{align*}\lim_{x\to\infty}\frac{\mathcal{Z}^{\mathsf{t}}(x)}{\rho^{\lfloor x\rfloor}(1+(\rho-1)\{x\})}=\lim_{x\to\infty}\frac{\mathcal{Z}^{\mathsf{t}}(x)}{\rho^{x}l_\rho(x)}=\frac{1}{\rho-1}W.\end{align*}

We recall that $l_{\rho}\;:\;[0,\infty)\to\mathbb{R}$ is defined as $l_{\rho}(x)=(1+(\rho-1)\{x\})\rho^{-\{x\}}$ . Since $\mathcal{Z}^{\mathsf{t}}(\tau_k)=k$ , we infer that for any $\delta>0$ and large enough k we have

\begin{align*}\frac{k}{\rho^{\tau_k}l_\rho(\tau_k)}\le \frac{1}{\rho-1}We^{\delta}\quad \text{and}\quad \frac{k}{\rho^{\tau_k-\delta}l_\rho(\tau_k-\delta)}\ge \frac{1}{\rho-1} We^{-\delta}. \end{align*}

This can be rewritten as

\begin{align*} \tau_k +\log_{\rho}l_{\rho}(\tau_k-\delta)-\log_{\rho}e^{-\delta}\leq T_k \leq \tau_k +\log_{\rho} l_{\rho}(\tau_k)+\log_{\rho}e^{\delta}. \end{align*}

Notice that we have

\begin{align*} \tau_k+\log_{\rho} l_{\rho}(\tau_k)= \lfloor \tau_k\rfloor + \log_{\rho}(1+(\rho-1)\{\tau_k\})=h^{-1}(\tau_k),\end{align*}

and the inverse of the increasing function $h^{-1}(x)$ is given by $ \lfloor x\rfloor +\frac{\rho^{\{x\}}-1}{\rho-1}=h(x)$ , which then yields the following inequalities:

\begin{align*}\lfloor T_k -\log_{\rho}e^{\delta}\rfloor+\dfrac{\rho^{\{T_k-\log_{\rho}e^{\delta}\}}-1}{\rho-1}\le\tau_k &\le \lfloor T_k -\log_{\rho}e^{-\delta}\rfloor+\dfrac{\rho^{\{T_k-\log_{\rho}e^{-\delta}\}}-1}{\rho-1}+\delta. \end{align*}

From the uniform continuity of h(x), by letting $\delta\to 0$ , we obtain the claim.

The above proposition implies that

\begin{align*}\tau_n=\log_{\rho}n+O(1),\quad\mathbb{P}^\mathcal{S}\text{-almost surely.}\end{align*}

4.4. Limit theorems for $B_j$

Proposition 6. Under the assumptions of Theorem 3, let $\Phi\;:\;\mathbb{Z}\to\mathbb{R}^{1\times J}$ be any characteristic such that the following stable convergence holds:

(21) \begin{align}\frac{\mathcal{X}^{\Phi}(n+x)}{n^{\ell+\frac12}\rho^{n/2}\sqrt W}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} \mathcal{G}^{\Phi}(x)\quad\text{in }\mathcal D(\mathbb{R}),\end{align}

for some continuous Gaussian process $\mathcal{G}^{\Phi}$ with $\textrm{Var} \left[\mathcal{G}^{\Phi}(x)\right]>0$ , for any $x\in\mathbb{R}$ . Then there exists a continuous, positive, 1-periodic function $\psi^{\Phi}$ such that for $\tau_n$ as in (10) and $T_n=\log_\rho\frac{n(\rho-1)}{W}$ , it holds that

(22) \begin{align} \frac{\mathcal{X}^{\Phi}(\tau_n)}{ \sqrt n (\log_\rho n)^{l+\frac12}\psi^{\Phi}(T_n) }\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} \mathcal N(0,1),\quad \text{ as } n\to\infty. \end{align}

Proof. The key idea in the proof is to use the functional limit theorem for $\mathcal{X}^{\Phi}$ to replace $\mathcal{X}^{\Phi}(\tau_n)$ by $\mathcal{X}^{\Phi}(t_n)$ . Recall that for h as defined in (20), which is continuous and strictly increasing, we have used the notation $t_n=h(\log_\rho n-\log_\rho\frac{W}{\rho-1})=h(T_n)$ . Furthermore, for any $x\in\mathbb{R}$ and $n\in\mathbb{N}$ it holds that $h(n+x)=n+h(x)$ . Consequently, by (21), we have

\begin{align*} \frac{\mathcal{X}^{\Phi} \big(h(n+x)\big)}{n^{\ell+\frac12}\rho^{n/2}\sqrt W}=\frac{\mathcal{X}^{\Phi} \big(n+h(x)\big)}{n^{\ell+\frac12}\rho^{n/2}\sqrt W}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} \mathcal{G}^{\Phi} \big(h(x)\big)\quad\text{in }\mathcal D(\mathbb{R}), \end{align*}

and the latter convergence can be rewritten as

\begin{align*} \frac{\mathcal{X}^{\Phi} \big(h(n+x)\big)}{\Big(n^{2\ell+1}\rho^{n} W\textrm{Var} \left[\mathcal{G}^{\Phi}\big(h(x)\big)\right]\Big)^{1/2}}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} G(x)\quad\text{in }\mathcal D(\mathbb{R}), \end{align*}

for some stationary and continuous process G with $G(0)\overset{\textrm{D}}{=}\mathcal N(0,1)$ . Note that one consequence of the convergence in (21) is the following property of the limiting process: $\mathcal{G}^{\Phi}(x+1)\overset{\textrm{d}}{=}\sqrt\rho \mathcal{G}^{\Phi}(x)$ . Hence, the previous convergence is equivalent to

\begin{align*} \frac{\mathcal{X}^{\Phi} \big(h(n+x)\big)}{\Big(\big((n+x)\vee 1\big)^{2\ell+1}\rho^{n+x}\rho^{-\{x\}}W\textrm{Var} \left[\mathcal{G}^{\Phi}\big(h(\{x\})\big)\right]\Big)^{1/2}}\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} G(x)\quad\text{in }\mathcal D(\mathbb{R}).\end{align*}

In other words, for the function $\psi^\Phi$ defined by

\begin{align*}\psi^\Phi(x)=\Big((\rho-1)\rho^{-\{x\}} \textrm{Var} \left[\mathcal{G}^{\Phi}\big(h(\{x\})\big)\right]\Big)^{1/2},\end{align*}

which is continuous and positive, and for

\begin{align*} X(x)=\frac{\mathcal{X}^{\Phi} \big(h(x)\big)}{\Big(\big(x\vee 1\big)^{2\ell+1}\rho^{x}\frac{W}{\rho-1}\Big)^{1/2} \psi^\Phi(x)}, \end{align*}

it holds that $X(n+x)\xrightarrow{\scriptscriptstyle{\textrm{st},\mathcal{S}}} G(x)$ , and therefore an application of Lemma 3(ii) with $a_n= \log_\rho n$ yields

(23) \begin{align} X\big(h^{-1}(t_n)\big)=X\big(\log_\rho n-\log_\rho\tfrac{W}{\rho-1}\big)\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} G(0).\end{align}

Since $h^{-1}(x)$ is uniformly continuous, by Proposition 5 we get

(24) \begin{align} h^{-1}(\tau_n)-h^{-1}(t_n)\to 0,\quad\mathbb{P}^\mathcal{S}\text{-almost surely}.\end{align}

We claim that from Lemma 3(i) with $N_n= \lfloor{\log_{\rho} n}\rfloor$ and $\delta_n=2^{-n}$ it follows that

(25) \begin{align} X\big(h^{-1}(\tau_n)\big)-X\big(h^{-1}(t_n)\big) \xrightarrow{\scriptscriptstyle{\mathbb{P}^\mathcal{S}}}0, \quad \text{as } n\to\infty.\end{align}

Indeed, for fixed $m\in\mathbb{N}$ , on the event $|\log_{\rho} W|\le k_m-2\delta_m-1-\log_\rho(\rho-1)$ and $|h^{-1}(\tau_n)-h^{-1}(t_n)|\le \delta_m$ we have

\begin{align*}\big|X\big(h^{-1}(\tau_n)\big)-X\big(h^{-1}(t_n)\big)\big|\le \omega(X_{N_{m}},k_m,\delta_{m}).\end{align*}

In turn, for any $\varepsilon>0$ we get

\begin{align*} &\mathbb{P}^\mathcal{S}\Big(\big|X\big(h^{-1}(\tau_n)\big)-X\big(h^{-1}(t_n)\big)\big|>\varepsilon\Big)\le \mathbb{P}^\mathcal{S}\big( \omega(X_{N_{m}},k_m,\delta_{m})>\varepsilon\big)\\[5pt] &+\mathbb{P}^\mathcal{S}\big(|\log_\rho W|> k_m-2\delta_m-1-\log_\rho(\rho-1)\big) +\mathbb{P}^\mathcal{S}\big(|h^{-1}(\tau_n)-h^{-1}(t_n)|> \delta_m\big).\end{align*}

Taking the limit first as $n\to\infty$ and then as $m\to\infty$ and using (24), we get (25). Finally, (25) and (23) imply that

\begin{align*}\frac{\mathcal{X}^{\Phi}(\tau_n)}{\big(h^{-1}(\tau_n)\vee 1\big)^{\ell+\frac12}\sqrt n\rho^{(h^{-1}(\tau_n)-h^{-1}(t_n))/2} \psi^\Phi(h^{-1}(\tau_n))}=X\big(h^{-1}(\tau_n)\big)\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} G(0),\end{align*}

which together with (24) and the $\mathbb{P}^\mathcal{S}$ -almost sure convergence of $\frac{\psi^\Phi(h^{-1}(\tau_n))}{\psi^\Phi(h^{-1}(t_n))}$ and $\frac{h^{-1}(\tau_n)}{\log_\rho n}$ to 1 yields that

\begin{align*} \frac{\mathcal{X}^{\Phi}(\tau_n)}{\sqrt n(\log_\rho n)^{\ell+\frac12} \psi^\Phi(T_n)}\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} \mathcal N(0,1);\end{align*}

that is, (22) holds. This completes the proof.

Using the auxiliary result that we have just proved, we can now state and prove our main result.

Theorem 4. Suppose that (15) holds and all assumptions from Theorem 3 are satisfied. Then there exists a continuous, positive, and 1-periodic function $\psi$ such that, for $T_n=\log_{\rho}\frac{n(\rho-1)}{W}$ and any $j\in[J]$ , we have

\begin{align*} \frac{B_j(n)-\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(n)\big)}{\sqrt n(\log_{\rho} n)^{\ell+\frac12}\psi(T_n)}\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} \mathcal{N}(0,1), \quad \text{as }n\to\infty. \end{align*}

Proof. Since for any $j\in[J]$ we have $\mathcal{Z}^{j}(\tau_n)=B_j(n)$ and $\mathcal{X}^{j}(n)=\mathcal{Z}^{j}(n)-\mathcal{F}^{j}(n)$ , we can write

\begin{align*}B_j(n)&=\mathcal{F}^{j}(\tau_n)+\mathcal{X}^{j}(\tau_n)\\[5pt] &=\mathcal{F}^{j}\big(\mathcal{F}^{\mathsf{inv}}(n)\big)+\mathcal{X}^{j}(\tau_n)+(\mathcal{F}^{j}\circ \mathcal{F}^{\mathsf{inv}}\circ \mathcal{F}^{\mathsf{t}}(\tau_n)-\mathcal{F}^{j}(\mathcal{F}^{\mathsf{inv}}(n))).\end{align*}

From the fact that $\mathcal{F}^{\mathsf{t}}(\tau_n)=n-\mathcal{X}^{\mathsf{t}}(\tau_n)$ together with Lemma 1, we obtain

\begin{align*}\mathcal{F}^{j}\circ \mathcal{F}^{\mathsf{inv}}\circ \mathcal{F}^{\mathsf{t}}(\tau_n)-\mathcal{F}^{j}\circ \mathcal{F}^{\mathsf{inv}}(n)+\rho \textrm{u}_j \mathcal{X}^{\mathsf{t}}(\tau_n)=\mathcal{X}^{\mathsf{t}}(\tau_n)o(1),\quad \mathbb{P}^\mathcal{S}\text{-almost surely},\end{align*}

and therefore

\begin{align*}B_j(n)-\mathcal{F}^{j}(\mathcal{F}^{\mathsf{inv}}(n))=\mathcal{X}^{j}(\tau_n)-\rho \textrm{u}_j\mathcal{X}^{\mathsf{t}}(\tau_n)+o(1)\mathcal{X}^{\mathsf{t}}(\tau_n)\quad \mathbb{P}^\mathcal{S}\text{-almost surely}.\end{align*}

By Proposition 4 (positivity of the variances of the limiting process) and Lemma 2 (continuity of the limit processes), the characteristics $\Phi=\Phi^j-\rho\textrm{u}_j\Phi^{\mathsf{t}}$ and $\Phi=\Phi^{\mathsf{t}}$ and the corresponding processes $\mathcal{X}^{\Phi}$ with these characteristics satisfy the assumptions of Proposition 6. Thus we can apply Proposition 6 to the processes $\mathcal{X}^{\Phi}=\mathcal{X}^{j}-\rho\textrm{u}_j\mathcal{X}^{\mathsf{t}}$ and to $\mathcal{X}^{\mathsf{t}}$ to obtain

\begin{align*}\frac{\mathcal{X}^{j}(\tau_n)-{\rho}\textrm{u}_j\mathcal{X}^{\mathsf{t}}(\tau_n)}{ \sqrt n (\log_\rho n)^{\ell+\frac12}\psi(T_n)} \xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}}\mathcal N(0,1),\end{align*}

for some function $\psi$ which is continuous, positive, and 1-periodic; moreover, $\frac{\mathcal{X}^{\mathsf{t}}(\tau_n)}{ \sqrt n (\log_\rho n)^{\ell+\frac12} }\cdot o(1) \xrightarrow{\scriptscriptstyle{\mathbb{P}}} 0$ as $n\to\infty$ , which completes the proof.

Finally, in view of Theorem 4, it suffices to find an expansion of $\mathcal{F}^{j}(\mathcal{F}^{\mathsf{inv}}(n))$ up to an error of order $o(n^{\log_\rho\gamma})$ to prove Theorem 2 ; the latter will then follow immediately from the next corollary.

Corollary 3. Under the assumptions of Theorem 4, suppose that all eigenvalues in $\Gamma$ are simple. Then the following hold:

1. (i) If $\gamma \gt \sqrt \rho$ , then for any $\lambda\in\Gamma$ there exist a 1-periodic, continuous function $f_{\lambda}\;:\;\mathbb{R}\to\mathbb{C}$ and a random variable $X_{\lambda}$ such that

   \begin{align*}B_j(n)=\rho\textrm{u}_j\cdot n+\sum_{\lambda\in \Gamma}n^{\log_{\rho}\lambda}f_{\lambda}(T_n)X_{\lambda}+o_{\mathbb{P}}\Big(n^{\log_{\rho}\gamma}\Big).\end{align*}

2. (ii) If $\gamma=\sqrt \rho$ , then there is a 1-periodic, continuous function $\psi\;:\;\mathbb{R}\to(0,\infty)$ such that the following convergence holds:

   \begin{align*}\frac{B_j(n)-\rho\textrm{u}_j\cdot n}{\sqrt n(\log_{\rho} n)^{\frac12}\psi(T_n)}\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} \mathcal{N}(0,1),\quad \text{as }n\to\infty.\end{align*}

3. (iii) If $\gamma<\sqrt \rho$ , then there is a 1-periodic, continuous function $\psi\;:\;\mathbb{R}\to(0,\infty)$ such that the following convergence holds:

   \begin{align*}\frac{B_j(n)-\rho\textrm{u}_j\cdot n}{\sqrt n\psi(T_n)}\xrightarrow{\scriptscriptstyle{\textrm{d},\mathcal{S}}} \mathcal{N}(0,1),\quad \text{as }n\to\infty.\end{align*}

Proof. We handle the case (i) in detail; the other two cases are identical, so we leave the details to the interested reader. If $\gamma>\sqrt \rho$ , then by (5) we have

\begin{align*} \mathcal{F}^{\Phi}(n)&= \textrm{x}_1(\Phi)A^n_1 W^{(1)}+\textrm{x}_2(\Phi)A^n_2Z_0 = \sum_{\lambda\in\Gamma\cup\{\rho\}}\lambda^n\textrm{x}_1(\Phi)\pi_{\lambda}W^{(1)}+o(\gamma^n)\\[5pt] &=\rho^n\textrm{x}_1(\Phi) W\textrm u+\sum_{\lambda\in\Gamma}\lambda^n\textrm{x}_1(\Phi) W_{\lambda} \textrm u^{\lambda}+o(\gamma^n),\end{align*}

where we define $W_{\lambda} \textrm u^{\lambda}=\pi_{\lambda} W^{(\lambda)}$ for some scalar random variable $W_{\lambda}$ (this can be done since $\pi_{\lambda}$ is a projection on the space spanned by the eigenvector $\textrm u^\lambda$ ). In particular, as $\mathcal{F}^{j}$ and $\mathcal{F}^{\mathsf{t}}$ are piecewise linear between consecutive integer arguments, we conclude that

\begin{align*} \mathcal{F}^{j}(x)&=\rho^xl_{\rho}(x)\textrm{x}_1(\Phi_{0}^j) W\textrm u+\sum_{\lambda\in\Gamma}\lambda^xl_{\lambda}(x)\textrm{x}_1(\Phi_0^j) W_\lambda \textrm u^{\lambda}+o(\gamma^x),\end{align*}

and taking into account that $\textrm{x}_1(\Phi_{0}^j)= \frac{\lambda}{\lambda-1}\textrm{e}_j^\top\pi_\lambda$ , we finally get

\begin{align*} \mathcal{F}^{j}(x)&=\frac{\rho^{x+1}}{\rho-1}l_\rho(x) W\textrm{u}_j+\sum_{\lambda\in\Gamma}\frac{\lambda^{x+1}}{\lambda-1}l_\lambda(x) W_\lambda \textrm{u}_j^\lambda+o(\gamma^x).\end{align*}

By the same argument, we also obtain

\begin{align*} \mathcal{F}^{\mathsf{t}}(x)=\sum_{j=1}^{J} \mathcal{F}^{j}(x-1)=\frac{\rho^x}{\rho-1}l_\rho(x) W+\sum_{\lambda\in\Gamma}\frac{\lambda^x}{\lambda-1}l_\lambda(x) W_\lambda \Big(\sum_{i=1}^{J}\textrm{u}_{i}^\lambda\Big) +o(\gamma^x).\end{align*}

In particular, it holds that

\begin{align*} \mathcal{F}^j(x)=\rho \textrm{u}_j\cdot \mathcal{F}^{\mathsf{t}}(x)+\sum_{\lambda\in\Gamma}\frac{\lambda^{x}}{\lambda-1}l_\lambda(x)\Big(\lambda\textrm{u}_j^\lambda-\rho \textrm u_j\Big(\sum_{i=1}^J\textrm{u}_i^\lambda\Big)\Big) W_\lambda+o(\gamma^x).\end{align*}

Since for $\lambda\in \Gamma$ we have

\begin{align*}\lambda^{x}=\bigg(\frac{\rho-1}{l_\rho(x) W}\mathcal{F}^{\mathsf{t}}(x)\bigg)^{\log_\rho \lambda}(1+o(1))=\bigg(\frac{\rho-1}{l_\rho(x) W}\mathcal{F}^{\mathsf{t}}(x)\bigg)^{\log_\rho \lambda}+o(\gamma^x) \text{ on } \mathcal{S},\end{align*}

and $\mathcal{F}^{\mathsf{inv}}(n)=t_n+o(1)=h(T_n)+o(1)$ , we deduce that

\begin{align*} \mathcal{F}^{j}(\mathcal{F}^{\mathsf{inv}}(n))& =\rho \textrm{u}_j\cdot n+\sum_{\lambda\in\Gamma}n^{\log_\rho \lambda} \frac{l_\lambda(h(T_n))}{l_\rho(h(T_n))^{\log_\rho \lambda}} \cdot\Big(\lambda\textrm{u}_j^\lambda-\rho \textrm u_j\Big(\sum_{i=1}^J \textrm{u}_i^\lambda\Big)\Big)\Big(\frac{\rho-1}{ W}\Big)^{\log_\rho \lambda} \frac{W_\lambda}{\lambda-1}\\[5pt] &+o_{\mathbb{P}}(n^{\log_\rho\gamma}),\end{align*}

and thus (i) holds with

\begin{align*}f_\lambda(x)=\frac{l_\lambda(h(x))}{l_\rho(h(x))^{\log_\rho \lambda}}\quad \text{ and} \quad X_\lambda= \big(\lambda\textrm{u}_j^\lambda-\rho \textrm u_j\big(\sum_{i=1}^J\textrm{u}_i^\lambda\big)\big)\Big(\frac{\rho-1}{ W}\Big)^{\log_\rho \lambda} \frac{W_\lambda}{\lambda-1},\end{align*}

for every $\lambda\in \Gamma$ .

In the case $\gamma=\sqrt \rho$ we have

\begin{align*} \mathcal{F}^{\Phi}(n)=\rho^n\textrm{x}_1(\Phi) W \textrm u +\sum_{\lambda\in\Gamma}\lambda^n\textrm{x}_1(\Phi) \pi_\lambda Z_0+o(\gamma^n)=\rho^n\textrm{x}_2(\Phi)W \textrm u +o(\rho^{n/2}),\end{align*}

and for $\gamma<\sqrt \rho$ we have

\begin{align*} \mathcal{F}^{\Phi}(n)=\rho^n\textrm{x}_1(\Phi) W \textrm u+o(\rho^{n/2}).\end{align*}

In both cases we can use the same approach as in the proof of (i), after an application of Theorem 4; this proves (ii) and (iii).