In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.

We study the existence of moments and the tail behavior of the densities of storage processes. We give sufficient conditions for existence and nonexistence of moments using the integrability conditions of submultiplicative functions with respect to Lévy measures. We then study the asymptotical behavior of the tails of these processes using the concave or convex envelope of the release rate function.

A new technique is developed for deriving velocities of propagation and shapes of travelling wave profiles in non-linear situations. Two models are considered for the spread of infection through a linear community. In one the infected individuals move, in the other the phenomenon of infection.

The parity of an individual in a population process is the number of offspring that individual has produced. If Nk(t) is the number of individuals of parity k at time t in a birth-and-death process and we show that, conditional on N(t)→ ∞,

In this paper, we consider Galton–Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.

We prove a functional law of large numbers and a functional central limit theorem for a controlled renewal process, that is, a point process which differs from an ordinary renewal process in that the ith interarrival time is scaled by a function of the number of previous i arrivals. The functional law of large numbers expresses the convergence of a sequence of suitably scaled controlled renewal processes to the solution of an ordinary differential equation. Likewise, the functional central limit theorem establishes that the error in the law of large numbers converges weakly to the solution of a stochastic differential equation. Our proofs are based on martingale and time-change arguments.

We define a new family of multivariate stochastic processes over a finite time horizon that we call generalised Liouville processes (GLPs). GLPs are Markov processes constructed by splitting Lévy random bridges into non-overlapping subprocesses via time changes. We show that the terminal values and the increments of GLPs have generalised multivariate Liouville distributions, justifying their name. We provide various other properties of GLPs and some examples.

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Leslie (1969) obtained the Laplace transform for the recurrence time of clusters of Poisson processes, which can be thought of as yielding the interspike interval distribution for a neuron that receives Poisson excitatory inputs subject to decay. Here, several extensions of this model are derived, each including Poisson inhibitory inputs. Expressions for the mean and variance are derived for each model, and the results for the different models are compared.

Using a well-known transformation that preserves Wiener measure a theorem is proved that enables one to derive in a non-computational manner distributions of simple functionals of the so-called adjusted range process.

The coalescent was introduced by Kingman (1982a), (1982b) and Tajima (1983) as a continuous-time Markov chain model describing the genealogical relationship among sampled genes from a panmictic population of a species. The random mating in a population is a strict condition and the genealogical structure of the population has a strong influence on the genetic variability and the evolution of the species. In this paper, starting from a discrete-time Markov chain model, we show the weak convergence to a continuous-time Markov chain, called the structured coalescent model, describing the genealogy of the sampled genes from whole population by means of passing the limit of the population size. Herbots (1997) proved the weak convergence to the structured coalescent on the condition of conservative migration and Wright–Fisher-type reproduction. We will give the proof on the condition of general migration rates and exchangeable reproduction.

As proposed by Irle and Gani in 2001, a process X is said to be slower in level crossing than a process Y if it takes X stochastically longer to exceed any given level than it does Y. In this paper, we extend a result of Irle (2003), relative to the level crossing ordering of uniformizable skip-free-to-the-right continuous-time Markov chains, to derive a new set of sufficient conditions for the level crossing ordering of these processes. We apply our findings to birth-death processes with and without catastrophes, and M/M/s/c systems.

An ergodic L2-theorem for inhomogeneous Markov chains covering simulated annealing with or without constraints and stochastic relaxation with or without constraints arising in Bayesian image reconstruction is proved. The derivation is self-contained.

In this paper, it is shown that the Foster-Lyapunov criterion is sufficient to ensure the existence of an invariant probability measure for both discrete- and continuous-time Markov processes without any additional hypotheses (such as irreducibility).

In this paper we study the behavior of a delayed compound renewal process, S, about some fixed level, L. Normally, a jump process S increases at random times τ1, τ2, …, in random increments until it crosses L. S would then be terminated in a random number v of phases at time τv. In many applications, a more general termination scenario assumes that S may evolve either through v or σ random phases, whichever of the two is smaller (denoted by T). The number T of actual phases is called the termination index, and we evaluate a joint functional of T, the termination time τ T and the termination level ST. We also seek information about the process S a step before its termination, and derive a joint functional for all relevant processes. Examples of these processes and their applications to various stochastic models are discussed.

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

We consider the inverse problem of estimating the input random field in a stochastic integral equation relating two random fields. The purpose of this paper is to present an approach to this problem using a Riesz-based or orthonormal-based series expansion of the input random field with uncorrelated random coefficients. We establish conditions under which the input series expansion induces (via the integral equation) a Riesz-based or orthonormal-based series expansion for the output random field. The estimation problem is studied considering two cases, depending on whether data are available from either the output random field alone, or from both the input and output random fields. Finally, we discuss this approach in the case of transmissivity estimation from piezometric head data, which was the original motivation of this work.

We prove a result concerning the joint distribution of alleles at linked loci on a chromosome drawn from the population at stationarity. For a neutral locus, the allele is a draw from the stationary distribution of the mutation process. Furthermore, this allele is independent of the alleles at different loci on any chromosomes in the population.