A Markov chain model related to population genetics and its convergence to a diffusion process on the multi-dimensional bounded domain are treated. We discuss the case where natural selection is random and the different selection effects over successive generations are independent. Our model is a multi-allelic version of the haploid model of Karlin and Levikson. The asymptotic properties of the limiting diffusion are stated.

Correction terms are obtained for the diffusion approximation to one- and two-barrier ruin problems in finite and infinite time. The corrections involve moments of ladder-height distributions, and a method is given for calculating them numerically. Examples show that the corrected approximations can be much more accurate than the originals.

In one version of the familiar ‘secretary problem’, n rankable individuals appear sequentially in random order, and a selection procedure (stopping rule) is found to minimize the expected rank of the individual selected. It is assumed here that, instead of being a fixed integer n, the total number of individuals present is a bounded random variable N, of known distribution. The form of the optimal stopping rule is given, and for N belonging to a certain class of distributions, depending on n, and such that E(N) → ∞ as n → ∞, some asymptotic results concerning the minimal expected rank are given.

Statistical inference for Markov processes is commonly based on the maximum likelihood method of estimation and the likelihood ratio criterion for testing hypotheses. Construction of estimators and test statistics by these methods require that a model be chosen in the form of a family of transition density functions. In this paper, asymptotic properties of the maximum likelihood estimator and of the likelihood ratio statistic λn are examined when the model chosen for their construction is incorrect—that is, when no density in the model is a density for the transition probability distribution of the Markov process. It is shown that if and λn are constructed from a ‘regular’ incorrect model, then is consistent and asymptotically normally distributed and the asymptotic null distribution of −2 log λn is that of a linear combination of independent chi-squared random variables. These results are applied to propose measures of the performance of the test based on λn when the statistic is constructed from an incorrect model.

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

Let t ≥ 0 be a Markov transition probability semigroup on a general space satisfying a suitable φ-irreducibility condition. We show the existence of (i) a decay parameter λ ≥ 0 which is a common abscissa of convergence of the integrals ƒ estPt (x, A) dt for almost all x and all suitable A, (ii) a natural classification into λ-positive, λ-null and λ-transient cases. Moreover this classification is completely determined by any one of the h-skeleton chains of (Pt). We study the convergence of eλtPt(x, A) in the λ-positive case, and show that the limit f(X)π(A) (where f and π are the unique λ-invariant function and measure, normalized so that π(f) = 1) is reached at a uniform exponential rate of convergence, i.e. ||eλtPt(x, ·)-f(x)π(·)||f = O (e−αt) for some α > 0 and almost all x if there is a π-positive set such that the convergence is exponentially fast on this set. These results are used to deduce conditions for (Pt) to have quasi-stationary distributions.

Suppose two alternative designs for a stochastic system are to be compared. These two systems can be simulated independently or dependently. This paper presents a method for comparing two regenerative stochastic processes in a dependent fashion using common random numbers. A set of sufficient conditions is given that guarantees that the dependent simulations will produce a variance reduction over independent simulations. Numerical examples for a variety of simple stochastic models are included which illustrate the variance reduction achieved.

Let denote the empirical distribution obtained from a sequence of i.i.d. -valued random vectors with common distribution P. If is a class of Borel subsets of then we say that it forms a Glivenko–Cantelli class for P if In this paper we describe a simple technique for identifying such classes, based on the idea of uniformity classes for setwise convergence. Classes for which the method proves successful include the closed half-spaces, closed balls, and the class of all convex subsets of .

Random closed sets (in Matheron's sense) which are a.s. compact convex and contain the origin are considered. The totality of such random closed sets are closed under the Minkowski addition and we can define the concept of infinite divisibility with respect to Minkowski addition of random compact convex sets. Using a generalized notion of Laplace transformations we get Lévy-type canonical representations of infinitely divisible random compact convex sets. Isotropic and stable cases are also considered. Finally we get several mean formulas of Minkowski functionals of infinitely divisible random compact convex sets in terms of their Lévy spectral measures.

Let Un be the time between the nth and (n + 1)th arrivals to a single-server queuing system, and Vn the nth arrival's service time. There are quite a few models in which {Un, Vn, n ≥ 1} is a regenerative sequence. In this paper, some light and heavy traffic limit theorems are proved solely under this assumption; some of the light traffic results, and all the heavy traffic results, are new for two such models treated earlier by the author; and all the results are new for the semi-Markov queuing model.

In the last three sections, the results are applied to a single-server queue whose input is the output of a G/G/1 queue functioning in light traffic.

A queueing model of M/G/1 type with controllable service time distribution is considered in this paper. Conditions on costs and service time distribution only are established such that there exists a monotone policy optimal relative to the average cost criterion. To prove this, it is shown that it suffices to search for a monotone optimal policy in a Markov decision model associated with the queueing model.