Corrections are made to formulations and proofs of some theorems about convolution equivalence closure for random sum distributions. These arise because of the falsity of a much used asymptotic equivalence lemma, and they impinge on the convolution equivalence closure theorem for general infinitely divisible laws.

In this paper we consider the problem of first-crossing from above for a partial sums process m+S t , t ≥ 1, with the diagonal line when the random variables X t , t ≥ 1, are independent but satisfying nonstationary laws. Specifically, the distributions of all the X t s belong to a common parametric family of arithmetic distributions, and this family of laws is assumed to be stable by convolution. The key result is that the first-crossing time distribution and the associated ballot-type formula rely on an underlying polynomial structure, called the generalized Abel-Gontcharoff structure. In practice, this property advantageously provides simple and efficient recursions for the numerical evaluation of the probabilities of interest. Several applications are then presented, for constant and variable parameters.

In this paper we study reliability properties of consecutive-k-out-of-n systems with exchangeable components. For 2k ≦ n, we show that the reliability functions of these systems can be written as negative mixtures (i.e. mixtures with some negative weights) of two series (or parallel) systems. Some monotonicity and asymptotic properties for the mean residual lifetime function are obtained and some ordering properties between these systems are established. We prove that, under some assumptions, the mean residual lifetime function of the consecutive-k-out-of-n: G system (i.e. a system that functions if and only if at least k consecutive components function) is asymptotically equivalent to that of a series system with k components. When the components are independent and identically distributed, we show that consecutive-k-out-of-n systems are ordered in the likelihood ratio order and, hence, in the mean residual lifetime order, for 2k ≦ n. However, we show that this is not necessarily true when the components are dependent.

We show that there exist symmetric properties in the discrete n-cube whose threshold widths range asymptotically between 1/√n and 1/logn. These properties are built using a combination of failure sets arising in reliability theory. This combination of sets is simply called a product. Some general results on the threshold width of the product of two sets A and B in terms of the threshold locations and widths of A and B are provided.

This paper presents a study of the intertemporal propagation of distributional properties of phenotypes in general polygenic multisex inheritance models with sex- and time-dependent heritabilities. It further analyzes the implications of these models under heavy-tailedness of traits' initial distributions. Our results suggest the optimality of a flexible asexual/binary mating system. Switching between asexual and binary inheritance mechanisms allows the population effectively to achieve a fast suppression of negative traits and a fast dispersion of positive traits, regardless of the distributional properties of the phenotypes in the initial period.

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.

Most devices (systems) are operated under different environmental conditions. The failure process of a system not only depends on the intrinsic characteristics of the system itself but also on the external environmental conditions under which the system is being operated. In this paper we study a stochastic failure model in a random environment and investigate the effect of the environmental factors on the failure process of the system.

Blackwell (1951), in his seminal work on comparison of experiments, ordered two experiments using a dilation ordering: one experiment, Y, is ‘more spread out’ in the sense of dilation than another one, X, if E(c(Y))≥E(c(X)) for all convex functions c. He showed that this ordering is equivalent to two other orderings, namely (i) a total time on test ordering and (ii) a martingale relationship E(Yʹ | Xʹ)=Xʹ, where (Xʹ,Yʹ) has a joint distribution with the same marginals as X and Y. These comparisons are generalized to balayage orderings that are defined in terms of generalized convex functions. These balayage orderings are equivalent to (i) iterated total integral of survival orderings and (ii) martingale-type orderings which we refer to as k-mart orderings. These comparisons can arise naturally in model fitting and data confidentiality contexts.

The distribution theory for reward functions on semi-Markov processes has been of interest since the early 1960s. The relevant asymptotic distribution theory has been satisfactorily developed. On the other hand, it has been noticed that it is difficult to find exact distribution results which lead to the effective computation of such distributions. Note that there is no satisfactory exact distribution result for rewards accumulated over deterministic time intervals [0, t], even in the special case of continuous-time Markov chains. The present paper provides neat general results which lead to explicit closed-form expressions for the relevant Laplace transforms of general reward functions on semi-Markov and Markov additive processes.

Suppose Xi, i = 1,…,n are indepedent and identically distributed with E/X1/r < ∞, r = 1,2,…. If Cov (( − μ)r, S2) = 0 for r = 1, 2,…, where μ = EX1, S2 = , and , then we show X1 ~ N (μ, σ2), where σ2 = Var(X1). This covariance zero condition charaterizes the normal distribution. It is a moment analogue, by an elementary approach, of the classical characterization of the normal distribution by independence of and S2 using semi invariants. More generally, if Cov = 0 for r = 1,…, k, then E((X1 − μ)/σ)r+2 = EZr+2 for r = 1,… k, where Z ~ N(0, 1). Conversely Corr may be arbitrarily close to unity in absolute value, but for unimodal X1, Corr2( < 15/16, and this bound is the best possible.

Let F(x ) denote a distribution function in R d and let F *n (x ) denote the nth convolution power of F(x ). In this paper we discuss the asymptotic behaviour of 1 - F *n (x ) as x tends to ∞ in a certain prescribed way. It turns out that in many cases 1 - F *n (x ) ∼ n(1 - F(x )). To obtain results of this type, we introduce and use a form of subexponential behaviour, thereby extending the notion of multivariate regular variation. We also discuss subordination, in which situation the index n is replaced by a random index N.

We study the genealogy of so-called immortal branching processes, i.e. branching processes where each individual upon death is replaced by at least one new individual, and conclude that their marginal distributions are compound geometric. The result also implies that the limiting distributions of properly scaled supercritical branching processes are compound geometric. We exemplify our results with an expression for the marginal distribution for a class of branching processes that have recently appeared in the theory of coalescent processes and continuous stable random trees. The limiting distribution can be expressed in terms of the Fox H-function, and in special cases by the Meijer G-function.

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

We characterize the (continuous) majorization of integrable functions introduced by Hardy, Littlewood, and Pólya in terms of the (discrete) majorization of finite-dimensional vectors, introduced by the same authors. The most interesting version of this result is the characterization of the (increasing) convex order for integrable random variables in terms of majorization of vectors of expected order statistics. Such a result includes, as particular cases, previous results by Barlow and Proschan and by Alzaid and Proschan, and, in a sense, completes the picture of known results on order statistics. Applications to other stochastic orders are also briefly considered.

Under some relaxed conditions, in this paper we obtain some equivalent conditions on the asymptotics of the density of the supremum of a random walk with heavy-tailed increments. To do this, we investigate the asymptotics of the first ascending ladder height of a random walk with heavy-tailed increments. The results obtained improve and extend the corresponding classical results.

In the present article we derive a new discrete multivariate distribution using a limiting argument that is essentially the same as the law of small numbers. The distribution derived belongs to an exponential family, and randomly partitions positive integers. The facts shown about the distribution are useful in many fields of application involved with count data. The derivation parallels that of the Ewens distribution from the gamma distribution, and the new distribution is produced from the inverse Gaussian distribution. The method employed is regarded as the discretization of an infinitely divisible distribution over nonnegative real numbers.

In this paper we consider stochastic recursive equations of sum type, , and of max type, , where A i , b i , and b are random, (X i ) are independent, identically distributed copies of X, and denotes equality in distribution. Equations of these types typically characterize limits in the probabilistic analysis of algorithms, in combinatorial optimization problems, and in many other problems having a recursive structure. We develop some new contraction properties of minimal L s -metrics which allow us to establish general existence and uniqueness results for solutions without imposing any moment conditions. As an application we obtain a one-to-one relationship between the set of solutions to the homogeneous equation and the set of solutions to the inhomogeneous equation, for sum- and max-type equations. We also give a stochastic interpretation of a recent transfer principle of Rösler from nonnegative solutions of sum type to those of max type, by means of random scaled Weibull distributions.

Let X = (X 1, X 2, …, X n ) be an exchangeable random vector, and write X (1:i) = min{X 1, X 2, …, X i }, 1 ≤ i ≤ n. In this paper we obtain conditions under which X (1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

Fractional Laplace motion is obtained by subordinating fractional Brownian motion to a gamma process. Used recently to model hydraulic conductivity fields in geophysics, it might also prove useful in modeling financial time series. Its one-dimensional distributions are scale mixtures of normal laws, where the stochastic variance has the generalized gamma distribution. These one-dimensional distributions are more peaked at the mode than is a Gaussian distribution, and their tails are heavier. In this paper we derive the basic properties of the process, including a new property called stochastic self-similarity. We also study the corresponding fractional Laplace noise, which may exhibit long-range dependence. Finally, we discuss practical methods for simulation.