Let X1, X2, · ·· be a sequence of independent, identically distributed (i.i.d.) random variables with positive mean. An analogue of Rényi's (1962) stochastic geyser problem is solved for the associated process of first-passage times. More precisely, it is shown that a single realization of the sequence determines the distribution function (d.f.) of the Xn's almost surely (a.s.), even if the observations are erroneous up to an order o(log n).

In this note, necessary and sufficient conditions for the central limit theorem for the number of events in a doubly stochastic Poisson process are given.

The paper studies a network of queues in which units arrive singly in a Poisson stream at a service channel S from which they branch out into k parallel channels S1, S2, …, Sk. After having been serviced at S1, S2, … Sk, the units converge again into a single channel S′. The service times of units at each of the channels are assumed to be exponential. Units finally serviced at S’ may leave the system or may again join S. This has been considered by taking two models denoted as Model A and Model B. Steady-state probabilities giving the number of units present in the system have been obtained explicitly for both the models. The expressions for mean queue lengths have also been arrived at.

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

In the type II counter with constant deadtime, particles which arrive within some constant time τ following another particle are unrecorded. We can think of this process as an alternating sequence of gaps and bunches of events. Gaps have duration > τ, while the intervals between any successive pair of events within a bunch are all ≦ τ. Counter theory is usually concerned with the distribution of intervals between recorded events (i.e., the first event of each bunch) and the distribution of the number of recorded events in a given time interval. In the case where the events form a renewal process this has been studied intensively by Pyke [2], Smith [5] and Takács [6].

Recent characterizations of probability measures by use of conditional expectations have included many probability measures defined on subsets of the real line. This paper contains results which allow similar characterizations by use of conditional expectations but are valid for probability measures defined on subsets of an arbitrary space. In the special case of a measure defined on subsets of R' the result is not always as sharp as known results, but these theorems can be applied to measures defined on subsets of Rn with n ≧ 1.

We describe a simple relation between the asymptotic behavior of the variance and of the expected number of distinct sites visited during a correlated random walk. The relation is valid for multistate random walks with finite variance in dimensions 1 and 2. A similar relation, valid in all dimensions, exists between the asymptotic behavior of the variance and of the probability of return to the origin.

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.

We extend the definition of level-crossing ordering of stochastic processes, proposed by Irle and Gani (2001), to the case in which the times to exceed levels are compared using an arbitrary stochastic order, and work, in particular, with integral stochastic orders closed for convolution. Using a sample-path approach, we establish level-crossing ordering results for the case in which the slower of the processes involved in the comparison is skip-free to the right. These results are specially useful in simulating processes that are ordered in level crossing, and extend results of Irle and Gani (2001), Irle (2003), and Ferreira and Pacheco (2005) for skip-free-to-the-right discrete-time Markov chains, semi-Markov processes, and continuous-time Markov chains, respectively.

We introduce a multivariate class of distributions with support I, a k-orthotope in $[0,\infty)^{k}$ , which is dense in the set of all k-dimensional distributions with support I. We call this new class ‘multivariate finite-support phase-type distributions’ (MFSPH). Though we generally define MFSPH distributions on any finite k-orthotope in $[0,\infty)^{k}$ , here we mainly deal with MFSPH distributions with support $[0,1)^{k}$ . The distribution function of an MFSPH variate is computed by using that of a variate in the MPH $^{*} $ class, the multivariate class of distributions introduced by Kulkarni (1989). The marginal distributions of MFSPH variates are found as FSPH distributions, the class studied by Ramaswami and Viswanath (2014). Some properties, including the mixture property, of MFSPH distributions are established. Estimates of the parameters of a particular class of bivariate finite-support phase-type distributions are found by using the expectation-maximization algorithm. Simulated samples are used to demonstrate how this class could be used as approximations for bivariate finite-support distributions.

In this paper we give a solution of an optimal stopping problem concerning random walks with non-zero drift, thereby proving the necessity of the existence of ESτ for Wald's equation ESτ = ES1 · Ετ to hold, even if attention is restricted to non-randomized stopping times τ. This answers a question of Robbins and Samuel (1966).

The rate of genetic drift at an autosomal locus for a bisexual, diploid population of fixed size is studied. The generations are non-overlapping. The model encompasses a variety of mating systems, including random monogamy, random polygamy in one sex and random mating. The rate of drift is shown for several models to depend on the expected number of parents that two randomly selected individuals have in common. The male and female offspring are assigned to families in a fairly general way, which permits the study of a model in which each family has offspring of one sex only. The equation arising in this last case is identical to one of Jacquard for a system in which sib-mating is excluded.

We consider s ≦ n randomly chosen permutations of the numbers 1, 2, …, n, and write them under each other, thus forming an s × n matrix, called “random-batch”. A rule, prescribing how many elements of a column may occur exactly one time, how many may occur exactly two times, etc., is called a fixpoint-structure. Assuming each possible permutation to be chosen with equal probability, the number of columns having a certain fixpoint-structure F is a random variable X(F). The limiting distribution of X(F) for n→ ∞ is considered for different cases of s = s(n). The main result (Theorem 4) says, that the common distribution of a finite number t of given fixpoint-structures tends to the product of t Poisson-laws, n→ ∞.

Beurling's inequality is proved to have an interpretation in terms of ingredients that do not depend on the effective numerical contribution of transition probabilities of circuit processes (those processes whose transition law is expressed by a denumerable class of directed weighted circuits). Connections with reversibility for Sk-state, k≧2, circuit processes, where S is a denumerable set, are revealed.

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

We consider percolation on the graph of the product of a regular tree T with degree d and the line ℤ, in which each tree edge is open with probability 1 – exp(–JTß) and each line edge is open with probability . Let C(o, 0) be the open cluster for . Denote by θ (β) the percolation probability. Here we show that θ (β) is continuous when ß > ßc, where ßc = sup{ß : θ(β) = 0}.

Feller (1968) showed that the probability functions of each of three random variables associated with coin-tossing (independent, simple, symmetric random walk) are (a) the same (except at the end points), (b) approximately arc-sine. In this paper these two properties are shown to hold for Markov-dependent coin-tossing (Markov-dependent, simple, symmetric random walk).

This paper aims at showing that for the discrete time analogue of the M/G/l queueing model with service in random order and with a traffic intensity ρ > 0, the condition ρ < ∞ is sufficient in order that every customer joining the queue be served eventually, with probability one (Theorem 2).

We consider epidemics in populations that are partitioned into small groups known as households. Whilst infectious, a typical infective makes global and local contact with individuals chosen independently and uniformly from the whole population or their own household, as appropriate. Previously, the classical Poisson approximation for the number of survivors of a severe epidemic has been extended to the household model. However, in the current work we exploit a Sellke-type construction of the epidemic process, which enables the derivation of sufficient conditions for the existence of a compound Poisson limit theorem for the survivors of the epidemic. The results are specialised to the Reed-Frost and general stochastic epidemic models.