The problem of finding bounds for P(A1 ∪ ⋯ ∪ An) based on P(Ak1 ∩ ⋯ ∩ Aki) (1 ≤ k1 < ⋯ < ki ≤ n, i = 1,…,d) goes back to Boole (1854), (1868) and Bonferroni (1937). In this paper upper bounds are presented using methods in graph theory. The main theorem is a common generalization of the earlier results of Hunter, Worsley and recent results of Prékopa and the author. Algorithms are given to compute bounds. Examples for bounding values of multivariate normal distribution functions are presented.

Let (Xn) be a sequence of independent, identically distributed random variables, with common distribution function F, possibly discontinuous. We use martingale arguments to connect the number of upper records from (Xn) with sums of minima of related random variables. From this relationship we derive a general strong law for the number of records for a wide class of distributions F, including geometric and Poisson.

A new computationally simple, speedy and accurate method is proposed to construct first-passage-time probability density functions for Gauss–Markov processes through time-dependent boundaries, both for fixed and for random initial states. Some applications to Brownian motion and to the Brownian bridge are then provided together with a comparison with some computational results by Durbin and by Daniels. Various closed-form results are also obtained for classes of boundaries that are intimately related to certain symmetries of the processes considered.

We analyse a non-Markovian generalization of the telegrapher's random process. It consists of a stochastic process describing a motion on the real line characterized by two alternating velocities with opposite directions, where the random times separating consecutive reversals of direction perform an alternating renewal process. In the case of Erlang-distributed interrenewal times, explicit expressions of the transition densities are obtained in terms of a suitable two-index pseudo-Bessel function. Some results on the distribution of the maximum of the process are also disclosed.

Multivariate subordinators are multivariate Lévy processes that are increasing in each component. Various examples of multivariate subordinators, of interest for applications, are given. Subordination of Lévy processes with independent components by multivariate subordinators is defined. Multiparameter Lévy processes and their subordination are introduced so that the subordinated processes are multivariate Lévy processes. The relations between the characteristic triplets involved are established. It is shown that operator self-decomposability and the operator version of the class Lm property are inherited from the multivariate subordinator to the subordinated process under the condition of operator stability of the subordinand.

This paper analyzes optimal single and multiple stopping rules for a class of correlated random walks that provides an elementary model for processes exhibiting momentum or directional reinforcement behavior. Explicit descriptions of optimal stopping rules are given in several interesting special cases with and without transaction costs. Numerical examples are presented comparing optimal strategies to simpler buy and hold strategies.

We define the extension of the so-called ‘martingales in the branching random walk’ in R or C to some Banach algebras B of infinite dimension and give conditions for their convergence, almost surely and in the Lp norm. This abstract approach gives conditions for the simultaneous convergence of uncountable families of such martingales constructed simultaneously in C, the idea being to consider such a family as a function-valued martingale in a Banach algebra of functions. The approach is an alternative to those of Biggins (1989), (1992) and Barral (2000), and it applies to a class of families to which the previous approach did not. We also give a result on the continuity of these multiplicative processes. Our results extend to a varying environment version of the usual construction: instead of attaching i.i.d. copies of a given random vector to the nodes of the tree ∪n≥0N+n, the distribution of the vector depends on the node in the multiplicative cascade. In this context, when B=R and in the nonnegative case, we generalize the measure on the boundary of the tree usually related to the construction; then we evaluate the dimension of this nonstatistically self-similar measure. In the self-similar case, our convergence results make it possible to simultaneously define uncountable families of such measures, and then to estimate their dimension simultaneously.

For a positive recurrent continuous-time Markov chain on a countable state space, we compare the access time to equilibrium to the hitting time of a particular state. For monotone processes, the exponential rates are ranked. When the process starts far from equilibrium, a cutoff phenomenon occurs at the same instant, in the sense that both the access time to equilibrium and the hitting time of a fixed state are equivalent to the expectation of the latter. In the case of Markov chains on trees, that expectation can be computed explicitly. The results are illustrated on the M/M/∞ queue.

The paper describes a new numerical method for the calculation of noncrossing probabilities for arbitrary boundaries by a Poisson process. We find the method to be simple in implementation, quick and efficient - it works reliably for Poisson processes of very high intensity n, up to several thousand. Hence, it can be used to detect unusual features in the finite-sample behaviour of empirical process and trace it down to very high sample sizes. It also can be used as a good approximation for noncrossing probabilities for Brownian motion and Brownian bridge, in particular when the boundaries are not regular. As a numerical example we demonstrate the divergence of normalized Kolmogorov-Smirnov statistics from their prescribed limiting distributions (Eicker (1979), Jaeshke (1979)) for quite large n in contrast to very regular behaviour of statistics of Mason (1983). For the Brownian motion case we considered square-root, Daniels' (1969) and Grooneboom's (1989) boundaries.

In this paper, we obtain Markovian bounds on a function of a homogeneous discrete time Markov chain. For deriving such bounds, we use well-known results on stochastic majorization of Markov chains and the Rogers–Pitman lumpability criterion. The proposed method of comparison between functions of Markov chains is not equivalent to generalized coupling method of Markov chains, although we obtain same kind of majorization. We derive necessary and sufficient conditions for existence of our Markovian bounds. We also discuss the choice of the geometric invariant related to the lumpability condition that we use.

The asymptotic behaviour of an occupation-time process associated with alternating renewal processes is investigated in the infinite mean cycle case. The limit theorems obtained extend some asymptotic results proved by Dynkin (1955), Lamperti (1958) and Erickson (1970) for the classical spent lifetime process. Some new phenomena are also presented.

In this paper, we study a maintenance model with general repair and two types of replacement: failure and preventive replacement. When the system fails a decision is made whether to replace or repair it. The repair degree that affects the virtual age of the system is assumed to be a random function of the repair-cost and the virtual age at failure time. The system can be preventively replaced at any time before failure. The objective is to find the repair/replacement policy minimizing the long-run expected average cost per unit time. It is shown that a generalized repair-cost-limit policy is optimal and the preventive replacement time depends on the virtual age of the system and on the length of the operating time since the last repair. Computational procedures for finding the optimal repair-cost limit and the optimal average cost are developed. This model includes many well-known models as special cases and the approach provides a unified treatment of a wide class of maintenance models.

This paper is about the probability law of the integral of geometric Brownian motion over a finite time interval. A partial differential equation is derived for the Laplace transform of the law of the reciprocal integral, and is shown to yield an expression for the density of the distribution. This expression has some advantages over the ones obtained previously, at least when the normalized drift of the Brownian motion is a non-negative integer. Bougerol's identity and a relationship between Brownian motions with opposite drifts may also be seen to be special cases of these results.

We prove that for a class of anisotropic long-range percolation models for which connection probabilities p<x,z> satisfy some regularity properties, and such that ∑z∈Z2p<x,z> = ∞, percolation still will occur even if we truncate all edges whose length exceeds some constant (which in this case depends on the family of connectivity probabilities {p<x,z>). We also present an example of dependent long-range percolation model for which this is not true.

We consider an M/G/1 queue with the special feature that the speed of the server alternates between two constant values sL and sH > sL. The high-speed periods are exponentially distributed, and the low-speed periods have a general distribution. Our main results are: (i) for the case that the distribution of the low-speed periods has a rational Laplace–Stieltjes transform, we obtain the joint distribution of the buffer content and the state of the server speed; (ii) for the case that the distribution of the low-speed periods and/or the service request distribution is regularly varying at infinity, we obtain explicit asymptotics for the tail of the buffer content distribution. The two cases in which the offered traffic load is smaller or larger than the low service speed are shown to result in completely different asymptotics.

Connections between classification and lumpability in the stochastic Hopfield model (SHM) are explored and developed. A simplification of the SHM's complexity based upon its inherent lumpability is derived. Contributions resulting from this reduction in complexity include: (i) computationally feasible classification time computations; (ii) a development of techniques for enumerating the stationary distribution of the SHM's energy function; and (iii) a characterization of the set of possible absorbing states of the Markov chain associated with the zero temperature SHM.

Some consequences of restarting stochastic search algorithms are studied. It is shown under reasonable conditions that restarting when certain patterns occur yields probabilities that the goal state has not been found by the nth epoch which converge to zero at least geometrically fast in n. These conditions are shown to hold for restarted simulated annealing employing a local generation matrix, a cooling schedule Tn ∼ c/n and restarting after a fixed number r + 1 of duplications of energy levels of states when r is sufficiently large. For simulated annealing with logarithmic cooling these probabilities cannot decrease to zero this fast. Numerical comparisons between restarted simulated annealing and several modern variations on simulated annealing are also presented and in all cases the former performs better.