The age and type distributions (conditioned on non-extinction) in a critical Bellman-Harris process are shown to converge in probability to a deterministic distribution under a finite first-moment hypothesis on the offspring distribution and a tightness condition on the lifetime distributions.

A recent paper in Advances in Applied Probability (Siegel (1976)) considered the duration of the time Tmn for a linear birth and death process to grow from a (large) initial size m to a larger size n. The main aim was to show that, when the birth rate exceeds the death rate, Tmn is close to its mean value, log n/m, with high probability. This paper establishes this result using much simpler techniques.

After some reminiscences about the founding of the Applied Probability journals the author discusses three open problems concerned respectively with:

(i) the prehistory of applied probability

(ii) Poisson–Delauney Simplexes

(iii) voids and pancakes in the universe.

Let {W(t), 0 ≦ t < ∞} be the standard Wiener process. The techniques of computing probabilities of the type are well known. The main purpose of this paper is to present ways of finding barrier-absorption probabilities when the barrier function is defined only on sub-intervals of [0, T].

Direct and simple proofs are given of the equivalence of partial balance over a set of states and the fact that the corresponding equilibrium distribution is insensitive to nominal sojourn time in that set and independent of its past and remaining duration. By a counter-example it is shown that solely insensitivity is not sufficient for partial balance.

The paper presents sufficient conditions for certain functions to be convex. Functions of this type often appear in Markov decision processes, where their maximum is the solution of the problem. Since a convex function takes its maximum at an extreme point, the conditions may greatly simplify a problem. In some cases a full solution may be obtained after the reduction is made. Some illustrative examples are discussed.

A method is presented for constructing a continuous, stationary process with a given rate of upcrossings at all levels and for which it is possible to find the exact distribution of the number of upcrossings, in an interval, at all levels.

Consider an independent site percolation model on Z d , with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1} N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We also show some results concerning how K(p) should scale with p as p goes to 0. Related results are also obtained for the question of whether or not almost all words are seen.

Upper deviation results are obtained for the split time of a supercritical continuous-time Markov branching process. More precisely, we establish the existence of logarithmic limits for the likelihood that the split times of the process are greater than an identified value and determine an expression for the limiting quantity. We also give an estimation for the lower deviation probability of the split times, which shows that the scaling is completely different from the upper deviations.

Consider a random vector $\textbf{U}$ whose distribution function coincides in its upper tail with that of an Archimedean copula. We report the fact that the conditional distribution of $\textbf{U}$ , conditional on one of its components, has under a mild condition on the generator function independent upper tails, no matter what the unconditional tail behavior is. This finding is extended to Archimax copulas.

We study a renewal theory approach to perpetual two-state switching problems with infinite value functions. Since the corresponding value functions are infinite, the problems fall outside the standard class of problems which can be analyzed using dynamic programming. Instead, we propose an alternative formulation of optimal switching theory in which optimality of a strategy is defined in terms of its long-term mean return, which can be determined using renewal theory. The approach is illustrated by examples in connection with trend-following strategies in finance.

The limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ,μ. For any λ>0 we consider the percolation threshold μc(λ) associated to the parameter μ. Denoting by λc the percolation threshold for the standard Poisson Boolean model with radii r, we show the lower bound μc(λ)≥clog(c∕(λ−λc)) for any λ>λc with c>0 a fixed constant. In particular, there is no phase transition in μ at the critical value of λ, that is, μc(λc) =∞.

We study an example of a hit-and-run random walk on the symmetric group $\mathbf S_n$ . Our starting point is the well-understood top-to-random shuffle. In the hit-and-run version, at each single step, after picking the point of insertion j uniformly at random in $\{1,\ldots,n\}$ , the top card is inserted in the jth position k times in a row, where k is uniform in $\{0,1,\ldots,j-1\}$ . The question is, does this accelerate mixing significantly or not? We show that, in $L^2$ and sup-norm, this accelerates mixing at most by a constant factor (independent of n). Analyzing this problem in total variation is an interesting open question. We show that, in general, hit-and-run random walks on finite groups have non-negative spectrum.

Log-concavity of a joint survival function is proposed as a model for bivariate increasing failure rate (BIFR) distributions. Its connections with or distinctness from other notions of BIFR are discussed. A necessary and sufficient condition for a bivariate survival function to be log-concave (BIFR-LCC) is given that elucidates the impact of dependence between lifetimes on ageing. Illustrative examples are provided to explain BIFR-LCC for both positive and negative dependence.

We consider solutions of Lévy-driven stochastic differential equations of the form $\textrm{d} X_t=\sigma(X_{t-})\textrm{d} L_t$, $X_0=x$, where the function $\sigma$ is twice continuously differentiable and the driving Lévy process $L=(L_t)_{t\geq0}$ is either vector or matrix valued. While the almost sure short-time behavior of Lévy processes is well known and can be characterized in terms of the characteristic triplet, there is no complete characterization of the behavior of the solution X. Using methods from stochastic calculus, we derive limiting results for stochastic integrals of the form $t^{-p}\int_{0+}^t\sigma(X_{t-})\,\textrm{d} L_t$ to show that the behavior of the quantity $t^{-p}(X_t-X_0)$ for $t\downarrow0$ almost surely reflects the behavior of $t^{-p}L_t$. Generalizing $t^{{\kern1pt}p}$ to a suitable function $f\colon[0,\infty)\rightarrow\mathbb{R}$ then yields a tool to derive explicit law of the iterated logarithm type results for the solution from the behavior of the driving Lévy process.

Let $(Z_n)_{n\geq0}$ be a supercritical Galton–Watson process. Consider the Lotka–Nagaev estimator for the offspring mean. In this paper we establish self-normalized Cramér-type moderate deviations and Berry–Esseen bounds for the Lotka–Nagaev estimator. The results are believed to be optimal or near-optimal.

Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes. Our results complement the results known in the literature for the subcases of Jackson networks with either infinite supply nodes or unstable nodes by providing an analysis of the significantly more challenging case of networks with both types of nonstandard node present. Building on our product-form results, we provide closed-form solutions for common customer and system oriented performance measures.

Pitman (2003), and subsequently Gnedin and Pitman (2006), showed that a large class of random partitions of the integers derived from a stable subordinator of index $\alpha\in(0,1)$ have infinite Gibbs (product) structure as a characterizing feature. The most notable case are random partitions derived from the two-parameter Poisson–Dirichlet distribution, $\textrm{PD}(\alpha,\theta)$, whose corresponding $\alpha$-diversity/local time have generalized Mittag–Leffler distributions, denoted by $\textrm{ML}(\alpha,\theta)$. Our aim in this work is to provide indications on the utility of the wider class of Gibbs partitions as it relates to a study of Riemann–Liouville fractional integrals and size-biased sampling, and in decompositions of special functions, and its potential use in the understanding of various constructions of more exotic processes. We provide characterizations of general laws associated with nested families of $\textrm{PD}(\alpha,\theta)$ mass partitions that are constructed from fragmentation operations described in Dong et al. (2014). These operations are known to be related in distribution to various constructions of discrete random trees/graphs in [n], and their scaling limits. A centerpiece of our work is results related to Mittag–Leffler functions, which play a key role in fractional calculus and are otherwise Laplace transforms of the $\textrm{ML}(\alpha,\theta)$ variables. Notably, this leads to an interpretation within the context of $\textrm{PD}(\alpha,\theta)$ laws conditioned on Poisson point process counts over intervals of scaled lengths of the $\alpha$-diversity.

Starting from a Markov chain with a finite or a countable infinite alphabet, we consider the chain obtained when all but one symbol are indistinguishable for the practitioner. We study conditions on the transition matrix of the Markov chain ensuring that the image chain has continuous or discontinuous transition probabilities with respect to the past.