Let (Yn, Nn)n≥1 be independent and identically distributed bivariate random variables such that the Nn are positive with finite mean ν and the Yn have a common heavy-tailed distribution F. We consider the process (Zn)n≥1 defined by Zn = Yn - Σn-1, where It is shown that the probability that the maximum M = maxn≥1Zn exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

Let $(\xi_k,\eta_k)_{k\in\mathbb{N}}$ be independent identically distributed random vectors with arbitrarily dependent positive components. We call a (globally) perturbed random walk a random sequence $T\,{:\!=}\, (T_k)_{k\in\mathbb{N}}$ defined by $T_k\,{:\!=}\, \xi_1+\cdots+\xi_{k-1}+\eta_k$ for $k\in\mathbb{N}$ . Consider a general branching process generated by T and let $N_j(t)$ denote the number of the jth generation individuals with birth times $\leq t$ . We treat early generations, that is, fixed generations j which do not depend on t. In this setting we prove counterparts for $\mathbb{E}N_j$ of the Blackwell theorem and the key renewal theorem, prove a strong law of large numbers for $N_j$ , and find the first-order asymptotics for the variance of $N_j$ . Also, we prove a functional limit theorem for the vector-valued process $(N_1(ut),\ldots, N_j(ut))_{u\geq0}$ , properly normalized and centered, as $t\to\infty$ . The limit is a vector-valued Gaussian process whose components are integrated Brownian motions.

We investigate the average similarity of random strings as captured by the average number of ‘cousins’ in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m-ary, and the corresponding trees are m-ary trees. Certain analytic issues arise in the m-ary case that do not have an analog in the binary case.

Based on pathwise duality constructions, several new results on truncated queues and storage systems of the G/M/1 type are derived by transforming the workload (content) processes into certain ‘dual’ M/G/1-type processes. We consider queueing systems in which (a) any service requirement that would increase the total workload beyond the capacity is truncated so as to keep the associated sojourn time below a certain constant, or (b) new arrivals do not enter the system if they have to wait more than one time unit in line. For these systems, we derive the steady-state distributions of the workload and the numbers of customers present in the systems as well as the distributions of the lengths of busy and idle periods. Moreover, we use the duality approach to study finite capacity storage systems with general state-dependent outflow rates. Here our duality leads to a Markovian finite storage system with state-dependent jump sizes whose content level process can be analyzed using level crossing techniques. We also derive a connection between the steady-state densities of the non-Markovian continuous-time content level process of the G/M/1 finite storage system with state-dependent outflow rule and the corresponding embedded sequence of peak points (local maxima).

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.

For many years the modified Mercalli (MM) scale has been used to describe earthquake damage and effects observed at scattered locations. In the next stage of an analysis involving MM data, isoseismal lines based on the observations have been added to maps by hand, i.e. subjectively. However a few objective methods have been proposed (by e.g. De Rubeis et al., Brillinger, Wald et al. and Pettenati et al.). The work presented here develops objective methods further. In particular the ordinal character of the MM scale is specifically taken into account. Numerical smoothing is basic to the approach and methods involving splines, local polynomial regression and wavelets are illustrated. The approach also allows the inclusion of explanatory variables, for example site effects. The procedure is implemented for data from the 17 October 1989 Loma Prieta earthquake.

Gorostiza and Wakolbinger (1991), and Dawson and Perkins (1991) established the same persistence criterion for a class of critical branching particle systems and for a class of superprocesses respectively. In this note we take another approach to the criterion and present a simpler proof of it.

A Markovian process with rewards is considered. The interference into its development consists in instantaneous changes of state called replacements. It is shown that, for any replacement policy not depending on the future, the expected reward is a linear function of the expected replacement numbers. The application of this result is illustrated by examples.

This paper is concerned with the optimal number of redundant allocation to n-component coherent systems consisting of heterogeneous dependent components. We assume that the system is built up of L groups of different components, $L\geq 1$ , where there are $n_i$ components in group i, and $\sum_{i=1}^{L}n_i=n$ . The problem of interest is to allocate $v_i$ active redundant components to each component of type i, $i=1,\dots, L$ . To get the optimal values of $v_i$ we propose two cost-based criteria. One of them is introduced based on the costs of renewing the failed components and the costs of refreshing the alive ones at the system failure time. The other criterion is proposed based on the costs of replacing the system at its failure time or at a predetermined time $\tau$ , whichever occurs first. The expressions for the proposed functions are derived using the mixture representation of the system reliability function based on the notion of survival signature. We assume that a given copula function models the dependency structure between the components. In the particular case that the system is a series-parallel structure, we provide the formulas for the proposed cost-based functions. The results are discussed numerically for some specific coherent systems.

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 general counting processes stopped at a random time T, independent of the process. Provided that T has the decreasing failure rate (DFR) property, we present sufficient conditions on the arrival times so that the number of events occurring before T preserves the DFR property of T. In particular, when the interarrival times are independent, we consider applications concerning the DFR property of the stationary number of customers waiting in queue for specific queueing models.

In this paper a stochastic failure model for a system with stochastically dependent competing failures is analyzed. The system is subject to two types of failure: degradation failure and catastrophic failure. Both types of failure share an initial common source: an external shock process. This implies that they are stochastically dependent. In our developments of the model, the type of dependency between the two kinds of failure will be characterized. Conditional properties of the two competing risks are also investigated. These properties are the fundamental basis for the development of the maintenance strategy studied in this paper. Considering this maintenance strategy, the long-run average cost rate is derived and the optimal maintenance policy is discussed.

We establish a functional large deviation principle and a functional moderate deviation principle for Markov-modulated risk models with reinsurance by constructing an exponential martingale approach. Lundberg's estimate of the ruin time is also presented.

Taylor’s power law (or fluctuation scaling) states that on comparable populations, the variance of each sample is approximately proportional to a power of the mean of the population. The law has been shown to hold by empirical observations in a broad class of disciplines including demography, biology, economics, physics, and mathematics. In particular, it has been observed in problems involving population dynamics, market trading, thermodynamics, and number theory. In applications, many authors consider panel data in order to obtain laws of large numbers. Essentially, we aim to consider ergodic behaviors without independence. We restrict our study to stationary time series, and develop different Taylor exponents in this setting. From a theoretical point of view, there has been a growing interest in the study of the behavior of such a phenomenon. Most of these works focused on the so-called static Taylor’s law related to independent samples. In this paper we introduce a dynamic Taylor’s law for dependent samples using self-normalized expressions involving Bernstein blocks. A central limit theorem (CLT) is proved under either weak dependence or strong mixing assumptions for the marginal process. The limit behavior of the estimation involves a series of covariances, unlike the classic framework where the limit behavior involves the marginal variance. We also provide an asymptotic result for a goodness-of-fit procedure suitable for checking whether the corresponding dynamic Taylor’s law holds in empirical studies.

We consider a multiclass single-server queueing network as a model of a packet switching network. The rates packets are sent into this network are controlled by queues which act as congestion windows. By considering a sequence of congestion controls, we analyse a sequence of stationary queueing networks. In this asymptotic regime, the service capacity of the network remains constant and the sequence of congestion controllers act to exploit the network's capacity by increasing the number of packets within the network. We show that the stationary throughput of routes on this sequence of networks converges to an allocation that maximises aggregate utility subject to the network's capacity constraints. To perform this analysis, we require that our utility functions satisfy an exponential concavity condition. This family of utilities includes weighted α-fair utilities for α > 1.

We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned with counts of the number of vertices of a particular containment level.

For the local containment profile, we obtain the exact mean, variance, and probability distribution in terms of standard combinatorial quantities such as generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution.

As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers. The orders in the variance–covariance matrix establish concentration laws.

A method for proving the invariance of hitting-time distributions with respect to the control strategy of a Markov chain is presented. The method is applied to resource-sharing problems. It provides a new proof of a known result and extends it. Sufficient conditions for such an invariance are given and are illustrated by examples.

A population of susceptible individuals partitioned into m groups and exposed to a contagious disease is considered.

The progress of this simple epidemic is modeled by an m -dimensional stochastic process, whose components are the number of infective individuals in the respective groups at time t. Exact and approximate formulas for the joint and marginal state probabilities are obtained.

A family of distribution-free statistics along with related tests is defined and properties of its members are studied. These statistics, one of which is the Smirnov-Wald and Wolfowitz statistic, Dn+ yield tests of the one-sided hypotheses. The minimax and maximin tests in this family against a restricted class of alternatives of minimum power are obtained. The connection to a confidence bound of Stringer for the mean is also remarked.