In this paper we assign a set of conditions to a strong Markov process and arrive at a differential equation analogous to the Kolmogorov equation. However, in this case the duration variable is the net distance travelled and the state variable is a time, a situation precisely opposite to that of Brownian motion. Solving this differential equation under certain boundary conditions produces the density function of the first passage times of Brownian motion with positive drift (see [1]), with the aid of which we define a new stochastic process.

By adding a vorticity matrix to the reversible transition probability matrix, we show that the commute time and average hitting time are smaller than that of the original reversible one. In particular, we give an affirmative answer to a conjecture of Aldous and Fill (2002). Further quantitive properties are also studied for the nonreversible finite Markov chains.

A Borel–Cantelli-type property in terms of one-step transition probabilities is given for events like {|Xn+1| > a + ε, |Xn|≦a}, a and ε being two positive numbers. Applications to normed sums of i.i.d. random variables with infinite mean and branching processes in varying environment with or without immigration are derived.

Suppose Q1⋆, … Qn⋆ are independent, identically distributed random variables, each with probability density function f(x), cumulative distribution function F(x), where F(1) – F(0) = 1, f(x) is continuous in the open interval (0, 1) and continuous on the right at x = 0 and on the left at x = 1, and there exists a positive C such that f(x) > C for all x in (0, l). f(0) is defined as f(0+), f(1) is defined as f(1–).

In a ballot, candidate A scores a votes and candidate B scores b votes. Suppose the ballots are drawn out one at a time, and denote αr and βr the number of votes registered for A and B, respectively, among the first r votes recorded. Further, let Δa,b be the number of subscripts r satisfying the strict lead condition , let be the number of subscripts r satisfying the weak lead condition ; and suppose all possible () voting records are equally probable. The probability distributions of the number of strict and weak lead positions corresponding to and , respectively, have been determined in [4] for a≧b.

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizes Mn vary without density dependence then the condition ΣMn–1 = ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.

Exact expressions are developed for the nth order autocovariance structure of the telegraph wave, the integral of which defines a biased correlated random walk.

The resolvent operators of the following two processes are obtained: (a) the bivariate Markov process W = (X, Y), where Y(t) is an irreducible Markov chain and X(t) is its integral, and (b) the geometric Wiener process G(t) = exp{B(t} where B(t) is a Wiener process with non-negative drift μ and variance parameter σ2. These results are then used via a limiting procedure to determine the long-run average cost per unit time of operating a finite dam where the input process is either X(t) or G(t). The system is controlled by a policy (Attia [1], Lam [6]).

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

Let Mn be the maximal spacing generated in the sample's range by Χ1,· ··, Χ n, independent and identically distributed Gaussian N(0, 1) random variables. We obtain the limiting distribution of and show that

according to whether ε> 0 or ε < 0.

The constancy of the conditional expectation of some appropriate functions of order statistics on some others, is used to characterize the geometric distribution among the discrete distributions.

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

A cascading failure is a failure in a system of interconnected parts in which the failure of a part can trigger the failure of successive parts. Although an initial and introductory approach for probabilistic modeling and analysis of the cascading failures was suggested in the literature, any general framework and fundamental results have yet to be reported. In this paper, applying the point process approach, we suggest a general framework for modeling and analysis of the cascading failures. Furthermore, a new concept of 'information-based residual lifetime' will be defined and discussed.

Inspired by methods of queueing theory, we propose a Markov model for the spread of viruses in an open population with an exogenous flow of infectives. We apply it to the diffusion of AIDS and hepatitis C diseases among drug users. From a mathematical point of view, the difference between the two viruses is shown in two parameters: the probability of curing the disease (which is 0 for AIDS but positive for hepatitis C) and the infection probability, which seems to be much higher for hepatitis. This model bears some resemblance to the M/M/∞ queueing system and is thus rather different from the models based on branching processes commonly used in the epidemiological literature. We carry out an asymptotic analysis (large initial population) and show that the Markov process is close to the solution of a nonlinear autonomous differential system. We prove both a law of large numbers and a functional central limit theorem to determine the speed of convergence towards the limiting system. The deterministic system itself converges, as time tends to ∞, to an equilibrium point. We then show that the sequence of stationary probabilities of the stochastic models shrinks to a Dirac measure at this point. This means that in a large population and for long-term analysis, we may replace the individual-based microscopic stochastic model with the macroscopic deterministic system without loss of precision. Moreover, we show how to compute the sensitivity of any functional of the Markov process with respect to a slight variation of any parameter of the model. This approach is applied to the spread of diseases among drug users, but could be applied to many other case studies in epidemiology.

Moderate deviation principles are established in dimensions d ≥ 3 for super-Brownian motion with random immigration, where the immigration rate is governed by the trajectory of another super-Brownian motion. It fills in the gap between the central limit theorem and large deviation principles for this model which were obtained by Hong and Li (1999) and Hong (2001).

Using operator methods, we derive a family of stochastic bounds for the Jackson network. For its transient joint queue-length distribution, we can stochastically bound it above by various networks that decouple into smaller independent Jackson networks. Each bound is determined by a distinct partitioning of the index set for the nodes. Except for the trivial cases, none of these bounds can be extended to a sample path ordering between it and the original network. Finally, we can partially order the bounds themselves whenever one partition of the index set is the refinement of another. These results suggest new types of partial orders for stochastic processes that are not equivalent to sample-path orderings.

A simple coupling argument is used to obtain a new proof of a result of Daniels (1967) concerning the total size distribution of the general stochastic epidemic. The proof admits a straightforward generalisation to multipopulation epidemics and indicates that similar results are unlikely to be available for epidemics with non-exponential infectious periods.

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.

A class of binary fission stochastic population models is described, in which the fission probabilities may depend on the age of an individual and the total population size. Age-dependent binary branching processes with Erlangian lifelength distributions are a special case. An asymptotic expression for the growth of the population size is developed, which generalizes known theorems about the asymptotic exponential growth of a branching process.