Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

This paper considers certain stochastic models for token and type counts in literary texts. Elaborating on some models of Gani, it is shown that reasonable fits can be obtained to some data of Yule on Macaulay's Essays.

Let Γn(p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where . Denote by γ n(p) the total number of cycles in the graph Γn(p). The main object of this paper is to prove that the limit distribution of γ n(λ /n) is a Poisson distribution with expectation if n → ∞and 0 < λ< 1.

This paper discusses the asymptotic behaviour, and in particular the critical singularities, of the generating functions for Pólya walks with a given fraction of visits to a hyperplane when the walk is or is not constrained to be on one side of that hyperplane. The three-dimensional case has applications in physics and chemistry.

We establish a sufficient condition for which the expected area under the trajectory of the carrier process is directly proportional to the expected number of removed carriers in the class of carrier-borne, right-shift, epidemic models studied by Severo (1969a). This result generalizes the previous work of Downton (1972) and Jerwood (1974) for some special cases of these models. We use the result to compute expected costs in the carrier-borne model due to Downton (1968) when it is unlikely that all the susceptibles will be infected. We conclude by showing that for the special case considered by Weiss (1965) this treatment of the expected cost is reasonable for populations with a large initial number of susceptibles.

An asymptotic expression is given for the expected number of vertices of the convex hull of a uniform sample from the interior of a d-dimensional simple polytope. This extends a result derived by Rényi and Sulanke for sample points in the plane.

We consider an infinite-capacity second-order fluid queue governed by a continuous-time Markov chain and with linear service rate. The variability of the traffic is modeled by a Brownian motion and a local variance function modulated by the Markov chain and proportional to the fluid level in the queue. The behavior of this second-order fluid-flow model is described by a linear stochastic differential equation, satisfied by the transient queue level. We study the transient level's convergence in distribution under weak assumptions and we obtain an expression for the stationary queue level. For the first-order case, we give a simple expression of all its moments as well as of its Laplace transform. For the second-order model we compute its first two moments.

In this paper, we analyse the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. For dimension greater than or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.

Several models for phage reproduction within a bacterium have recently been discussed by Steinberg and Stahl (1961), Gani (1962) and Ohlsen (1963); all are effectively birth-death type processes, with the first of these considering a single type of phage mutation. A detailed description of the biological processes involved may be found in Adams (1959), Weidel (1959), Sanders (1961) and Smith (1961). In this paper we shall consider some general models extending that of Steinberg and Stahl in which mutation may take place from normal a phages to mutant A phages.

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

Large deviation results are obtained for the normed limit of a supercritical multitype branching process. Starting from a single individual of type i, let L[i] be the normed limit of the branching process, and let be the minimum possible population size at generation k. If is bounded in k (bounded minimum growth), then we show that P(L[i] ≤ x) = P(L[i] = 0) + x α F *[i](x) + o(x α ) as x → 0. If grows exponentially in k (exponential minimum growth), then we show that −log P(L[i] ≤ x) = x −β/(1−β) G*[i](x) + o (x −β/(1−β)) as x → 0. If the maximum family size is bounded, then −log P(L[i] > x) = x δ/(δ−1) H *[i](x) + o(x δ/(δ−1)) as x → ∞. Here α, β and δ are constants obtained from combinations of the minimum, maximum and mean growth rates, and F *, G * and H * are multiplicatively periodic functions.

This paper presents a unified treatment of the convergence properties of nonhomogeneous Markov systems under different sets of assumptions. First the periodic case is studied and the limiting evolution of the individual cyclically moving subclasses of the state space of the associated Markov replacement chain is completely determined. A special case of the above result is the aperiodic or strongly ergodic convergence. Two numerical examples from the literature on manpower planning highlight the practical aspect of the theoretical results.

Let T be a non-empty subset of η X n stochastic matrices. Define The sequence T1, T2, · ·· is called a Markov set-chain. An important problem in this area is to determine when such a set-chain converges. This paper gives a notion of a sequential limiting set and shows how it can be used to obtain a result on set-chain convergence.

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

Consider an urn model whose replacement matrix is triangular, has all nonnegative entries, and the row sums are all equal to 1. We obtain strong laws for the counts of balls corresponding to each color. The scalings for these laws depend on the diagonal elements of a rearranged replacement matrix. We use these strong laws to study further behavior of certain three-color urn models.

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

Consider a semi-Markov process X(t) defined on a subset of the non-negative integers with zero as an absorbing state and the non-zero states forming an irreducible class with exit to zero being possible. Conditions are given for the existence of the limits: where Xj(t) is the amount of time prior to time t spent in state j.

The limits (which are independent of the initial state) are evaluated when the sufficient conditions are satisfied.

The resolution of the optimal stopping problem for a partially observed Markov state process reduces to the computation of a function — the Snell envelope — defined on a measure space which is in general infinite-dimensional. To avoid these computational difficulties, we propose in this paper to approximate the optimal stopping time as the limit of times associated to similar problems for a sequence of processes converging towards the true state. We show on two examples that these approximating states can be chosen such that the Snell envelopes can be explicitly computed.

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.