Probability generating functions are used to relate the joint distribution of the numbers of customers left behind by two successive departing customers to the marginal distribution of the number left behind by each departing customer. A probability generating function is then found for the joint distribution of the numbers of customers arriving in two successive departure intervals using the joint distribution of the numbers of customers left behind by three successive departing customers. The results could be obtained from general Markov chain theory but the method used in this paper is quicker.

Asymptotic formulas and Laplace–Stieltjes transforms are derived for the first two moments of a renewal process with a random number of delays. These are simplified when all the delays follow the same distribution. An asymptotic occupancy result is also derived for two-stage renewal processes with random numbers of delays. As an example, a demographic model of conception and birth is discussed. This model represents the sequence of live births to a woman as a renewal process. If the woman practises birth control after achieving her desired family composition, the renewal process has a random number of delays.

In a coverage problem introduced by Dvoretzky (1956) the circle is covered with lengths The associated indexed martingale can be considered as a set of random densities with respect to Lebesgue measure, of which the limit is a random measure. The total masses of the set constitute a new martingale. From a theorem of Shepp (1972), this martingale does not converge in a space if . If 0 < α < 1 it converges in all spaces , and in this paper we demonstrate that it converges in all spaces Lp for p > 2. We also obtain quite precise estimates for the moments of the total mass of the limit measure, showing that the characteristic function of this total mass is an entire function of order 1/(1 − α). Thus in some sense the mass is distributed in a bounded domain.

We introduce a stochastic process with discrete time and countable state space that is governed by a sequence of Markov matrices . Each Pm is used for a random number of steps Tm and is then replaced by Pm+1. Tm is a randomized stopping time that may depend on the most recent part of the state history. Thus the global character of the process is non-Markovian.

This process can be used to model the well-known simulated annealing optimization algorithm with randomized, partly state depending cooling schedules. Generalizing the concept of strong stationary times (Aldous and Diaconis [1]) we are able to show the existence of optimal schedules and to prove some desirable properties. This result is mainly of theoretical interest as the proofs do not yield an explicit algorithm to construct the optimal schedules.

Several information measures have been proposed and studied in the literature. One such measure is extropy, a complementary dual function of entropy. Its meaning and related aging notions have not yet been studied in great detail. In this paper, we first illustrate that extropy information ranks the uniformity of a wide array of absolutely continuous families. We then discuss several theoretical merits of extropy. We also provide a closed-form expression of it for finite mixture distributions. Finally, the dynamic versions of extropy are also discussed, specifically the residual extropy and past extropy measures.

We study the shape of the probability mass function of the Markov binomial distribution, and give necessary and sufficient conditions for the probability mass function to be unimodal, bimodal, or trimodal. These are useful to analyze the double-peaking results of a reactive transport model from the engineering literature. Moreover, we give a closed-form expression for the variance of the Markov binomial distribution, and expressions for the mean and the variance conditioned on the state at time n.

Measures of uncertainty are a topic of considerable and growing interest. Recently, the introduction of extropy as a measure of uncertainty, dual to Shannon entropy, has opened up interest in new aspects of the subject. Since there are many versions of entropy, a unified formulation has been introduced to work with all of them in an easy way. Here we consider the possibility of defining a unified formulation for extropy by introducing a measure depending on two parameters. For particular choices of parameters, this measure provides the well-known formulations of extropy. Moreover, the unified formulation of extropy is also analyzed in the context of the Dempster–Shafer theory of evidence, and an application to classification problems is given.

This paper proposes a Laplace-transform-based approach to price the fixed-strike quantile options as well as to calculate the associated hedging parameters (delta and gamma) under a hyperexponential jump diffusion model, which can be viewed as a generalization of the well-known Black-Scholes model and Kou's double exponential jump diffusion model. By establishing a relationship between floating- and fixed-strike quantile option prices, we can also apply this pricing and hedging method to floating-strike quantile options. Numerical experiments demonstrate that our pricing and hedging method is fast, stable, and accurate.

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix M and an element w of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.

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.

Suppose that an unknown number of objects arrive sequentially according to a Poisson process with random intensity λ on some fixed time interval [0,T]. We assume a gamma prior density Gλ(r, 1/a) for λ. Furthermore, we suppose that all arriving objects can be ranked uniquely among all preceding arrivals. Exactly one object can be selected. Our aim is to find a stopping time (selection time) which maximizes the time during which the selected object will stay relatively best. Our main result is the following. It is optimal to select the ith object that is relatively best and arrives at some time si(r) onwards. The value of si(r) can be obtained for each r and i as the unique root of a deterministic equation.

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.