We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

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.

Consider a sequence {X k , k ≥ 1} of random variables on (−∞, ∞). Results on the asymptotic tail probabilities of the quantities , and S (n) = max0 ≤ k ≤ n S k , with X 0 = 0 and n ≥ 1, are well known in the case where the random variables are independent with a heavy-tailed (subexponential) distribution. In this paper we investigate the validity of these results under more general assumptions. We consider extensions under the assumptions of having long-tailed distributions (the class L) and having the class D ∩ L, where D is the class of distribution functions with dominatedly varying tails. Some results are also given in the case where X k , k ≥ 1, are not necessarily identically distributed and/or independent.

Dependence structures for bivariate extremal events are analyzed using particular types of copula. Weak convergence results for copulas along the lines of the Pickands-Balkema-de Haan theorem provide limiting dependence structures for bivariate tail events. A characterization of these limiting copulas is also provided by means of invariance properties. The results obtained are applied to the credit risk area, where, for intensity-based default models, stress scenario dependence structures for widely traded products such as credit default swap baskets or first-to-default contract types are proposed.

Mixtures of increasing failure rate distributions can decrease, at least in some intervals of time. Usually this property is observed asymptotically, as t → ∞, which is due to the fact that a mixture failure rate is ‘bent down’, as the weakest populations are dying out first. We consider a survival model that generalizes additive hazards models, proportional hazards models, and accelerated life models very well known in reliability and survival analysis. We obtain new explicit asymptotic relations for a general setting and study specific cases. Under reasonable assumptions we prove that the asymptotic behavior of the mixture failure rate depends only on the behavior of the mixing distribution in the neighborhood of the left-hand endpoint of its support, and not on the whole mixing distribution.

In DNA sequences, specific words may take on biological functions as marker or signalling sequences. These may often be identified by frequent-word analyses as being particularly abundant. Accurate statistics is needed to assess the statistical significance of these word frequencies. The set of shuffled sequences - letter sequences having the same k-word composition, for some choice of k, as the sequence being analysed - is considered the most appropriate sample space for analysing word counts. However, little is known about these word counts. Here we present exact formulae for word counts in shuffled sequences.

We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

The conditional tail expectation in risk analysis describes the expected amount of risk that can be experienced given that a potential risk exceeds a threshold value, and provides an important measure of right-tail risk. In this paper, we study the convolution and extreme values of dependent risks that follow a multivariate phase-type distribution, and derive explicit formulae for several conditional tail expectations of the convolution and extreme values for such dependent risks. Utilizing the underlying Markovian property of these distributions, our method not only provides structural insight, but also yields some new distributional properties of multivariate phase-type distributions.

We provide sufficient conditions for the following types of random variable to have the increasing-failure-rate (IFR) property: sums of a random number of random variables; the time at which a Markov chain crosses a random threshold; the time until a random number of events have occurred in an inhomogeneous Poisson process; and the number of events of a renewal process, and of a general counting process, that have occurred by a randomly distributed time.

In this article, we consider the limit behavior of the hazard rate function of mixture distributions, assuming knowledge of the behavior of each individual distribution. We show that the asymptotic baseline function of the hazard rate function is preserved under mixture.

We study the asymptotic tail behavior of the conditional probability distributions of r t+k and r t+1+⋯+r t+k when (r t )t∈ℕ is a GARCH(1, 1) process. As an application, we examine the relation between the extreme lower quantiles of these random variables.

Throw n points sequentially and at random onto a unit circle and append a clockwise arc (or rod) of length s to each such point. The resulting random set (the free gas of rods) is a union of a random number of clusters with random sizes modelling a free deposition process on a one-dimensional substrate. A variant of this model is investigated in order to take into account the role of the disorder, θ > 0; this involves Dirichlet(θ) distributions. For such free deposition processes with disorder θ, we shall be interested in the occurrence times and probabilities, as n grows, of two specific types of configurations: those avoiding overlapping rods (the hard-rod gas) and those for which the largest gap is smaller than the rod length s (the packing gas). Special attention is paid to the thermodynamic limit when ns = ρ for some finite density ρ of points. The occurrence of parking configurations, those for which hard-rod and packing constraints are both fulfilled, is then studied. Finally, some aspects of these problems are investigated in the low-disorder limit θ ↓ 0 as n ↑ ∞ while nθ = γ > 0. Here, Poisson-Dirichlet(γ) partitions play some role.

In bioinformatics, the notion of an ‘island’ enhances the efficient simulation of gapped local alignment statistics. This paper generalizes several results relevant to gapless local alignment statistics from one to higher dimensions, with a particular eye to applications in gapped alignment statistics. For example, reversal of paths (rather than of discrete time) generalizes a distributional equality, from queueing theory, between the Lindley (local sum) and maximum processes. Systematic investigation of an ‘ownership’ relationship among vertices in ℤ2 formalizes the notion of an island as a set of vertices having a common owner. Predictably, islands possess some stochastic ordering and spatial averaging properties. Moreover, however, the average number of vertices in a subcritical stationary island is 1, generalizing a theorem of Kac about stationary point processes. The generalization leads to alternative ways of simulating some island statistics.

One approach to the computation of the price of an Asian option involves the Hartman–Watson distribution. However, numerical problems for its density occur for small values. This motivates the asymptotic study of its distribution function.

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k 1, k 2) denote the number of times that k 1 failures are followed by k 2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k 1, k 2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for N k3 (n; k 1, k 2), the number of times that k 1 failures followed by k 2 successes occur k 3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

This paper investigates the finite- and infinite-time ruin probabilities in a discrete-time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly-varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, and so apply in the case of infinite-time ruin.

Classes of life distributions based on the moment-generating-function order are investigated in this paper. It is shown firstly that the class ℳ is closed under both convex linear combination and geometric compounding. Secondly, the class NBUmg (new better than used in the moment-generating-function order) is proved to be closed under increasing star-shaped transformations. Finally, the interplay between the stochastic comparison of the excess lifetime of a renewal process and the NBUmg interarrivals is studied.

A method is provided for numerical evaluation, with any given accuracy, of the probability that at least p% of the genetic material from an individual's chromosomal segment survives to the next generation. Relevant MAPLE® V codes, for automated implementation of such evaluation, are also provided. The genomic continuum model, with Haldane's model for the crossover process, is assumed.

Let {X k , k ≥ 1} be a sequence of independently and identically distributed random variables with common subexponential distribution function concentrated on (−∞, ∞), and let τ be a nonnegative and integer-valued random variable with a finite mean and which is independent of the sequence {X k , k ≥ 1}. This paper investigates asymptotic behavior of the tail probabilities P(· > x) and the local probabilities P(x < · ≤ x + h) of the quantities and for n ≥ 1, and their randomized versions X (τ), S τ and S (τ), where X 0 = 0 by convention and h > 0 is arbitrarily fixed.

An asymptotic distribution is given for the partial sums of a stationary time-series with long-range dependence. The law of large numbers for the sample covariance of the series is also derived. The results differ from those given elsewhere in relaxing the assumption of the independence of the innovations of the series.