Let f be the density function associated to a matrix-exponential distribution of parameters $(\boldsymbol{\alpha}, T,\boldsymbol{{s}})$ . By exponentially tilting f, we find a probabilistic interpretation which generalizes the one associated to phase-type distributions. More specifically, we show that for any sufficiently large $\lambda\ge 0$ , the function $x\mapsto \left(\int_0^\infty e^{-\lambda s}f(s)\textrm{d} s\right)^{-1}e^{-\lambda x}f(x)$ can be described in terms of a finite-state Markov jump process whose generator is tied to T. Finally, we show how to revert the exponential tilting in order to assign a probabilistic interpretation to f itself.

We present a study of the joint distribution of both the state of a level-dependent quasi-birth–death (QBD) process and its associated running maximum level, at a fixed time t: more specifically, we derive expressions for the Laplace transforms of transition functions that contain this information, and the expressions we derive contain familiar constructs from the classical theory of QBD processes. Indeed, one important takeaway from our results is that the distribution of the running maximum level of a level-dependent QBD process can be studied using results that are highly analogous to the more well-established theory of level-dependent QBD processes that focuses primarily on the joint distribution of the level and phase. We also explain how our methods naturally extend to the study of level-dependent Markov processes of M/G/1 type, if we instead keep track of the running minimum level instead of the running maximum level.

Nonlinear Markov chains with finite state space were introduced by Kolokoltsov (Nonlinear Markov Processes and Kinetic Equations, 2010). The characteristic property of these processes is that the transition probabilities depend not only on the state, but also on the distribution of the process. Here we provide first results regarding their invariant distributions and long-term behaviour: we show that under a continuity assumption an invariant distribution exists and provide a sufficient criterion for the uniqueness of the invariant distribution. Moreover, we present examples of peculiar limit behaviour that cannot occur for classical linear Markov chains. Finally, we present for the case of small state spaces sufficient (and easy-to-verify) criteria for the ergodicity of the process.

Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

It was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing, and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monotonicity results associated with a supermodular function of two coordinates of our Markov process, from which the above-mentioned monotonicity and convexity/concavity results directly follow, but now for the class of Markov processes considered rather than just reflected Lévy processes. In addition, various results for the transient case (when the Markov process is not in stationarity) are provided. The conditions imposed are natural, in that they are satisfied by various frequently used Markovian models, as illustrated by a series of examples.

We analyse an additive-increase and multiplicative-decrease (also known as growth–collapse) process that grows linearly in time and that, at Poisson epochs, experiences downward jumps that are (deterministically) proportional to its present position. For this process, and also for its reflected versions, we consider one- and two-sided exit problems that concern the identification of the laws of exit times from fixed intervals and half-lines. All proofs are based on a unified first-step analysis approach at the first jump epoch, which allows us to give explicit, yet involved, formulas for their Laplace transforms. All eight Laplace transforms can be described in terms of two so-called scale functions associated with the upward one-sided exit time and with the upward two-sided exit time. All other Laplace transforms can be obtained from the above scale functions by taking limits, derivatives, integrals, and combinations of these.

We consider an (R, Q) inventory model with two types of orders, normal orders and emergency orders, which are issued at different inventory levels. These orders are delivered after exponentially distributed lead times. In between deliveries, the inventory level decreases in a state-dependent way, according to a release rate function $\alpha({\cdot})$ . This function represents the fluid demand rate; it could be controlled by a system manager via price adaptations. We determine the mean number of downcrossings $\theta(x)$ of any level x in one regenerative cycle, and use it to obtain the steady-state density f (x) of the inventory level. We also derive the rates of occurrence of normal deliveries and of emergency deliveries, and the steady-state probability of having zero inventory.

Considering a representative agent in the market, we study the long-term optimal investment problem in a discrete-time financial market, introducing a set of restrictions in the admissible strategies. The drawdown constraints limit the size of possible losses of the portfolio and impose a floor-based performance measure. The optimal growth rate is characterized, and under suitable hypotheses it is proved that an optimal strategy exists. The approach to solving this problem is based on dynamic programming techniques and a fixed point argument adapted from the theory of Markov decision processes.

This paper addresses the asymptotic analysis of sojourn functionals of spatiotemporal Gaussian random fields with long-range dependence (LRD) in time, also known as long memory. Specifically, reduction theorems are derived for local functionals of nonlinear transformation of such fields, with Hermite rank $m\geq 1,$ under general covariance structures. These results are proven to hold, in particular, for a family of nonseparable covariance structures belonging to the Gneiting class. For $m=2,$ under separability of the spatiotemporal covariance function in space and time, the properly normalized Minkowski functional, involving the modulus of a Gaussian random field, converges in distribution to the Rosenblatt-type limiting distribution for a suitable range of values of the long-memory parameter.

We introduce a general two-colour interacting urn model on a finite directed graph, where each urn at a node reinforces all the urns in its out-neighbours according to a fixed, non-negative, and balanced reinforcement matrix. We show that the fraction of balls of either colour converges almost surely to a deterministic limit if either the reinforcement is not of Pólya type or the graph is such that every vertex with non-zero in-degree can be reached from some vertex with zero in-degree. We also obtain joint central limit theorems with appropriate scalings. Furthermore, in the remaining case when there are no vertices with zero in-degree and the reinforcement is of Pólya type, we restrict our analysis to a regular graph and show that the fraction of balls of either colour converges almost surely to a finite random limit, which is the same across all the urns.

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

We present a recurrence–transience classification for discrete-time Markov chains on manifolds with negative curvature. Our classification depends only on geometric quantities associated to the increments of the chain, defined via the Riemannian exponential map. We deduce that a recurrent chain that has zero average drift at every point cannot be uniformly elliptic, unlike in the Euclidean case. We also give natural examples of zero-drift recurrent chains on negatively curved manifolds, including on a stochastically incomplete manifold.

This article derives quantitative limit theorems for multivariate Poisson and Poisson process approximations. Employing the solution of the Stein equation for Poisson random variables, we obtain an explicit bound for the multivariate Poisson approximation of random vectors in the Wasserstein distance. The bound is then utilized in the context of point processes to provide a Poisson process approximation result in terms of a new metric called $d_\pi$ , stronger than the total variation distance, defined as the supremum over all Wasserstein distances between random vectors obtained by evaluating the point processes on arbitrary collections of disjoint sets. As applications, the multivariate Poisson approximation of the sum of m-dependent Bernoulli random vectors, the Poisson process approximation of point processes of U-statistic structure, and the Poisson process approximation of point processes with Papangelou intensity are considered. Our bounds in $d_\pi$ are as good as those already available in the literature.

In this paper we study a class of optimal stopping problems under g-expectation, that is, the cost function is described by the solution of backward stochastic differential equations (BSDEs). Primarily, we assume that the reward process is $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable with $\mu>\mu_0$ for some critical value $\mu_0$ . This integrability is weaker than $L^p$ -integrability for any $p>1$ , so it covers a comparatively wide class of optimal stopping problems. To reach our goal, we introduce a class of reflected backward stochastic differential equations (RBSDEs) with $L\exp\bigl(\mu\sqrt{2\log\!(1+L)}\bigr)$ -integrable parameters. We prove the existence, uniqueness, and comparison theorem for these RBSDEs under Lipschitz-type assumptions on the coefficients. This allows us to characterize the value function of our optimal stopping problem as the unique solution of such RBSDEs.

We present a Markov chain on the n-dimensional hypercube $\{0,1\}^n$ which satisfies $t_{{\rm mix}}^{(n)}(\varepsilon) = n[1 + o(1)]$ . This Markov chain alternates between random and deterministic moves, and we prove that the chain has a cutoff with a window of size at most $O(n^{0.5+\delta})$ , where $\delta>0$ . The deterministic moves correspond to a linear shift register.

We study large-deviation probabilities of Telecom processes appearing as limits in a critical regime of the infinite-source Poisson model elaborated by I. Kaj and M. Taqqu. We examine three different regimes of large deviations (LD) depending on the deviation level. A Telecom process $(Y_t)_{t \ge 0}$ scales as $t^{1/\gamma}$ , where t denotes time and $\gamma\in(1,2)$ is the key parameter of Y. We must distinguish moderate LD ${\mathbb P}(Y_t\ge y_t)$ with $t^{1/\gamma} \ll y_t \ll t$ , intermediate LD with $ y_t \approx t$ , and ultralarge LD with $ y_t \gg t$ . The results we obtain essentially depend on another parameter of Y, namely the resource distribution. We solve completely the cases of moderate and intermediate LD (the latter being the most technical one), whereas the ultralarge deviation asymptotics is found for the case of regularly varying distribution tails. In all the cases considered, the large-deviation level is essentially reached by the minimal necessary number of ‘service processes’.

The study of threshold functions has a long history in random graph theory. It is known that the thresholds for minimum degree k, k-connectivity, as well as k-robustness coincide for a binomial random graph. In this paper we consider an inhomogeneous random graph model, which is obtained by including each possible edge independently with an individual probability. Based on an intuitive concept of neighborhood density, we show two sufficient conditions guaranteeing k-connectivity and k-robustness, respectively, which are asymptotically equivalent. Our framework sheds some light on extending uniform threshold values in homogeneous random graphs to threshold landscapes in inhomogeneous random graphs.

Scale-free percolation is a stochastic model for complex networks. In this spatial random graph model, vertices $x,y\in\mathbb{Z}^d$ are linked by an edge with probability depending on independent and identically distributed vertex weights and the Euclidean distance $|x-y|$ . Depending on the various parameters involved, we get a rich phase diagram. We study graph distance and compare it to the Euclidean distance of the vertices. Our main attention is on a regime where graph distances are (poly-)logarithmic in the Euclidean distance. We obtain improved bounds on the logarithmic exponents. In the light tail regime, the correct exponent is identified.

We study the geometric and topological features of U-statistics of order k when the k-tuples satisfying geometric and topological constraints do not occur frequently. Using appropriate scaling, we establish the convergence of U-statistics in vague topology, while the structure of a non-degenerate limit measure is also revealed. Our general result shows various limit theorems for geometric and topological statistics, including persistent Betti numbers of Čech complexes, the volume of simplices, a functional of the Morse critical points, and values of the min-type distance function. The required vague convergence can be obtained as a result of the limit theorem for point processes induced by U-statistics. The latter convergence particularly occurs in the $\mathcal M_0$ -topology.

Given a finite strongly connected directed graph $G=(V, E)$ , we study a Markov chain taking values on the space of probability measures on V. The chain, motivated by biological applications in the context of stochastic population dynamics, is characterized by transitions between states that respect the structure superimposed by E: mass (probability) can only be moved between neighbors in G. We provide conditions for the ergodicity of the chain. In a simple, symmetric case, we fully characterize the invariant probability.