We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernández and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events.

A correspondence formula between the laws of dual Markov chains on Z with two transition jumps is established. This formula contributes to the study of random walks in stationary random environments. Counterexamples with more than two jumps are exhibited.

Let X = {X_t: t ≥ 0} be a stationary piecewise continuous R^d-valued process that moves between jumps along the integral curves of a given continuous vector field, and let S ⊂ R^d be a smooth surface. The aim of this paper is to derive a multivariate version of Rice's formula, relating the intensity of the point process of (localized) continuous crossings of S by X to the distribution of X₀. Our result is illustrated by examples relating to queueing networks and stress release network models.

The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.

In the study of the reliability of technical systems in reliability engineering, coherent systems play a key role. In this paper we consider a coherent system consisting of n components with independent and identically distributed components and propose two time-dependent criteria. The first criterion is a measure of the residual lifetime of live components of a coherent system having some of the components alive when the system fails at time t. The second criterion is a time-dependent measure which enables us to investigate the inactivity times of the failed components of a coherent system still functioning though some of its components have failed. Several ageing and stochastic properties of the proposed measures are then established.

Almost all populations existing in the real world are finite populations. Specifically, in the areas relevant to lifetime modeling and analysis, finite populations are frequently encountered. However, descriptions of failure/survival patterns of elements in the finite population have not yet been properly established. In particular, it is questionable whether the ordinary failure rate can be defined for finite populations in the same way and whether the corresponding interpretations are still valid. In this paper we consider two kinds of finite mixed population and provide new definitions for their failure rates. Then we clarify the notion of failure rate in finite populations.

The notion of the signature is a basic concept and a powerful tool in the analysis of networks and reliability systems of binary type. An appropriate definition of this concept has recently been introduced for systems that have ν possible states (with ν ≥ 3). In this paper we analyze in detail several properties and the most relevant aspects of such a general definition. For simplicity's sake, we focus our attention on the case ν = 3. Our analysis will however provide a number of hints for understanding the basic aspects of the general case.

The concept of complete mixability is relevant to some problems of optimal couplings with important applications in quantitative risk management. In this paper we prove new properties of the set of completely mixable distributions, including a completeness and a decomposition theorem. We also show that distributions with a concave density and radially symmetric distributions are completely mixable.

We construct a process with inverse gamma increments and an asymptotically self-similar limit. This construction supports the use of long-range-dependent t subordinator models for actual financial data as advocated in Heyde and Leonenko (2005), in that it allows for noninteger-valued model parameters, as is found empirically in model estimation from data.

A positive recurrent, aperiodic Markov chain is said to be long-range dependent (LRD) when the indicator function of a particular state is LRD. This happens if and only if the return time distribution for that state has infinite variance. We investigate the question of whether other instantaneous functions of the Markov chain also inherit this property. We provide conditions under which the function has the same degree of long-range dependence as the chain itself. We illustrate our results through three examples in diverse fields: queueing networks, source compression, and finance.

Many natural populations are well modelled through time-inhomogeneous stochastic processes. Such processes have been analysed in the physical sciences using a method based on Lie algebras, but this methodology is not widely used for models with ecological, medical, and social applications. In this paper we present the Lie algebraic method, and apply it to three biologically well-motivated examples. The result of this is a solution form that is often highly computationally advantageous.

We present the construction of a continuous-time stochastic process which has moments that satisfy an exact scaling relation, including odd-order moments. It is based on a natural extension of the multifractal random walk construction described in Bacry and Muzy (2003). This allows us to propose a continuous-time model for the price of a financial asset that reflects most major stylized facts observed on real data, including asymmetry and multifractal scaling.

In this paper we study the asymptotic behavior of a general class of product-form closed queueing networks as the population size grows large. We first characterize the asymptotic behavior of the normalization constant for the stationary distribution of the network in exact order. This result then enables us to establish the asymptotic behavior of the system performance metrics, which extends a number of well-known asymptotic results to exact order. We further derive new, computationally simple approximations for performance metrics that significantly improve upon existing approximations for large-scale networks. In addition to their direct use for the analysis of large networks, these new approximations are particularly useful for reformulating large-scale queueing network optimization problems into more easily solvable forms, which we demonstrate with an optimal capacity planning example.

A discrete-time SIS model is presented that allows individuals in the population to vary in terms of their susceptibility to infection and their rate of recovery. This model is a generalisation of the metapopulation model presented in McVinish and Pollett (2010). The main result of the paper is a central limit theorem showing that fluctuations in the proportion of infected individuals around the limiting proportion converges to a Gaussian random variable when appropriately rescaled. In contrast to the case where there is no variation amongst individuals, the limiting Gaussian distribution has a nonzero mean.

In this paper we consider optimal stopping problems for a general class of reward functions under matrix-exponential jump-diffusion processes. Given an American call-type reward function in this class, following the averaging problem approach (see, for example, Alili and Kyprianou (2005), Kyprianou and Surya (2005), Novikov and Shiryaev (2007), and Surya (2007)), we give an explicit formula for solutions of the corresponding averaging problem. Based on this explicit formula, we obtain the optimal level and the value function for American call-type optimal stopping problems.

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

We study the dynamic assignment of cross-trained workers in serial production lines characterized by stochastic process times and inventory buffers between stations. Throughput maximization is the objective. Each worker is trained for a subset of tasks, where emphasis is placed on systems with each worker trained for a zone of stations with stations near the zone boundaries being served (shared) by one or more other workers as well. Using sample path comparisons, we identify structural properties of optimal worker allocation policies. We identify when (i) a worker can prioritize the job in the most downstream station (last-buffer-first-served), and (ii) only the downstream (as opposed to upstream) server should serve a single task.

A widely used model of carcinogenesis assumes that cells must go through a process of acquiring several mutations before they become cancerous. This implies that at any time there will be several populations of cells at different stages of mutation. In this paper we give exact expressions for the expectations and variances of the number of cells in each stage of such a stochastic multistage cancer model .