In the usual shock models, the shocks arrive from a single source. Bozbulut and Eryilmaz [(2020). Generalized extreme shock models and their applications. Communications in Statistics – Simulation and Computation 49(1): 110–120] introduced two types of extreme shock models when the shocks arrive from one of $m\geq 1$ possible sources. In Model 1, the shocks arrive from different sources over time. In Model 2, initially, the shocks randomly come from one of $m$ sources, and shocks continue to arrive from the same source. In this paper, we prove that the lifetime of Model 1 is less than Model 2 in the usual stochastic ordering. We further show that if the inter-arrival times of shocks have increasing failure rate distributions, then the usual stochastic ordering can be generalized to the hazard rate ordering. We study the stochastic behavior of the lifetime of Model 2 with respect to the severity of shocks using the notion of majorization. We apply the new stochastic ordering results to show that the age replacement policy under Model 1 is more costly than Model 2.

Log-concave random variables and their various properties play an increasingly important role in probability, statistics, and other fields. For a distribution F, denote by 𝒟F the set of distributions G such that the convolution of F and G has a log-concave probability mass function or probability density function. In this paper, we investigate sufficient and necessary conditions under which 𝒟F ⊆ 𝒟G, where F and G belong to a parametric family of distributions. Both discrete and continuous settings are considered.

In the reliability context, the geometric distribution is a natural choice to model the lifetimes of some equipment and components when they are measured by the number of completed cycles of operation or strokes, or in case of periodic monitoring of continuous data. This paper aims at investigating how the heterogeneity among the parameters affects some characteristics such as the distribution and hazard rate functions of spacings arising from independent heterogeneous geometric random variables. First, refined representations of the distribution functions are provided for both the second spacing and sample range from heterogeneous geometric sample. Second, stochastic comparisons are carried out on the second spacings and sample ranges for two sets of independent and heterogeneous geometric random variables in the sense of the usual stochastic and hazard rate orderings. The results established here not only fill the gap on the study of stochastic properties of spacings from heterogeneous geometric samples, but also are expected to be applied in the fields of statistics and reliability.

Let Xi,i ∈ ℕ, be independent and identically distributed random variables with values in ℕ0. We transform (‘prune’) the sequence {X1,…,Xn},n∈ ℕ, of discrete random samples into a sequence {0,1,2,…,Yn}, n∈ ℕ, of contiguous random sets by replacing Xn+1 with Yn +1 if Xn+1 >Yn. We consider the asymptotic behaviour of Yn as n→∞. Applications include path growth in digital search trees and the number of tables in Pitman's Chinese restaurant process if the latter is conditioned on its limit value.

Let (Xi)i∈ℕ be a sequence of independent and identically distributed random variables with values in the set ℕ0 of nonnegative integers. Motivated by applications in enumerative combinatorics and analysis of algorithms we investigate the number of gaps and the length of the longest gap in the set {X1,…,Xn} of the first n values. We obtain necessary and sufficient conditions in terms of the tail sequence (qk)k∈ℕ0, qk=P(X1≥ k), for the gaps to vanish asymptotically as n→∞: these are ∑k=0∞qk+1/qk <∞ and limk→∞qk+1/qk=0 for convergence almost surely and convergence in probability, respectively. We further show that the length of the longest gap tends to ∞ in probability if qk+1/qk→ 1. For the family of geometric distributions, which can be regarded as the borderline case between the light-tailed and the heavy-tailed situations and which is also of particular interest in applications, we study the distribution of the length of the longest gap, using a construction based on the Sukhatme–Rényi representation of exponential order statistics to resolve the asymptotic distributional periodicities.

The authors review various facts about copulas linking discrete distributions. They show how the possibility of ties that results from atoms in the probability distribution invalidates various familiar relations that lie at the root of copula theory in the continuous case. They highlight some of the dangers and limitations of an undiscriminating transposition of modeling and inference practices from the continuous setting into the discrete one.

We consider a sequence, of random length M, of independent, continuous observations Xi, 1 ≤ i ≤ M, where M is geometric, X1 has cumulative distribution function (CDF) G, and Xi, i ≥ 2, have CDF F. Let N be the number of upper records and let Rn, n ≥ 1, be the nth record value. We show that N is independent of F if and only if G(x) = G0(F(x)) for some CDF G0, and that if E(|X2|) is finite then so is E(|Rn|), n ≥ 2, whenever N ≥ n or N = n. We prove that the distribution of N, along with appropriately chosen subsequences of E(Rn), characterize F and G and, along with subsequences of E(Rn - Rn-1), characterize F and G up to a common location shift. We discuss some applications to the identification of the wage offer distribution in job search models.

We present several functional inequalitiesfor finite difference gradients, such asa Cheeger inequality, Poincaré and (modified) logarithmic Sobolev inequalities,associated deviation estimates,and an exponential integrability property.In the particular case of the geometric distribution on ${\mathbb{N}}$ we use an integration by parts formula to computethe optimal isoperimetric and Poincaré constants,and to obtain an improvement of ourgeneral logarithmic Sobolev inequality.By a limiting procedure we recover the correspondinginequalities for the exponential distribution.These results have applications to interacting spin systems undera geometric reference measure.

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

We consider a random record model from a continuous parent X with cumulative distribution function F, where the number of available observations is geometrically distributed. We show that, if E(|X|) is finite, then so is E(|Rn|) whenever Rn, the nth upper record value, exists. We prove that appropriately chosen subsequences of E(Rn) characterize F and subsequences of E(Rn − Rn−1) identify F up to a location shift. We discuss some applications to the identification of wage-offer distributions in job search models.

Miyazawa and Taylor (1997) introduced a class of assemble-transfer batch service queueing networks which do not have tractable stationary distribution. However by assuming a certain additional arrival process at each node when it is empty, they obtain a geometric product-form stationary distribution which is a stochastic upper bound for the stationary distribution of the original network. In this paper we develop a stochastic lower bound for the original network by introducing an additional departure process at each node which tends to remove all the customers present in it. This model in combination with the aforementioned upper bound model gives a better sense for the properties of the original network.

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

We introduce a batch service discipline, called assemble-transfer batch service, for continuous-time open queueing networks with batch movements. Under this service discipline a requested number of customers is simultaneously served at a node, and transferred to another node as, possibly, a batch of different size, if there are sufficient customers there; the node is emptied otherwise. We assume a Markovian setting for the arrival process, service times and routing, where batch sizes are generally distributed.

Under the assumption that extra batches arrive while nodes are empty, and under a stability condition, it is shown that the stationary distribution of the queue length has a geometric product form over the nodes if and only if certain conditions are satisfied for the extra arrivals. This gives a new class of queueing networks which have tractable stationary distributions, and simultaneously shows that the product form provides a stochastic upper bound for the stationary distribution of the corresponding queueing network without the extra arrivals.

We investigate the stationarity of minification processes when the marginal is a discrete distribution. There is a close relationship between the problem considered by Arnold and Isaacson (1976) and the stationarity in minification processes. We give a necessary and sufficient condition for a discrete distribution to be the marginal of a stationary minification process. Members of the Poisson and negative binomial families can be the marginals of stationary minification processes. The geometric minification process is studied in detail, and two characterizations of it based on the structure of the innovation process are given.

This paper considers the histogram of unit cell size built up from m independent observations on a Poisson (μ) distribution. The following question is addressed: what is the limiting probability of the event that there are no unoccupied cells lying to the left of occupied cells of the histogram? It is shown that the probability of there being no such isolated empty cells (or isolated finite groups of empty cells) tends to unity as the number m of observations tends to infinity, but that the corresponding almost sure convergence fails. Moreover this probability does not tend to unity when the Poisson distribution is replaced by the negative binomial distribution arising when μ is randomized by a gamma distribution. The relevance to empirical Bayes statistical methods is discussed.

Let γ t and δ t denote the residual life at t and current life at t, respectively, of a renewal process , with the sequence of interarrival times. We prove that, given a function G, under mild conditions, as long as holds for a single positive integer n, then is a Poisson process. On the other hand, for a delayed renewal process with the residual life at t, we find that for some fixed positive integer n, if is independent of t, then is an arbitrarily delayed Poisson process. We also give some corresponding results about characterizing the common distribution function F of the interarrival times to be geometric when F is discrete. Finally, we obtain some characterization results based on the total life or independence of γ t and δ t.

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c2, c3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

Maximum stability of a distribution with respect to a positive integer random variable N is defined by the property that the type of distribution is not changed when considering the maximum value of N independent observations. The logistic distribution is proved to be the only symmetric distribution which is maximum stable with respect to each member of a sequence of positive integer random variables assuming value 1 with probability tending to 1. If a distribution is maximum stable with respect to such a sequence and minimum stable with respect to another, then it must be logistic, loglogistic or ‘backward' loglogistic. The only possible sample size distributions in these cases are geometric.

Some consequences of a modified repair system for Phillips' (1981a, b) model for a two-component system are discussed. In the original model, both components are repaired whenever a revealed fault occurs; in the modified model only faulty components are repaired. Specifically (i) the distribution of time from the initial state up to discovery of an unrevealed fault, (ii) the expected proportion of time during which there exists an unrepaired fault, and (iii) the distribution of number of revealed faults up to and including the one which leads to a discovery of an unrevealed fault, are obtained. The theory is illustrated by examples, based on specific distributions for the times between repairs and occurrences of the two types of faults. A characterization of the exponential distribution is indicated.

This note gives a method of finding an upper bound for the mean of the maximum of n identically distributed non-negative random variables. The bound is explicitly given and numerically compared with the exact value of the mean of the maximum for some classical distributions (geometric, Poisson, Erlang, hyperexponential).