Asymptotic distributional properties of the maximal length aligned word (a contiguous set of letters) among multiple random Markov dependent sequences composed of letters from a finite alphabet are given. For sequences of length N, Cr,s(N) defined as the longest common aligned word found in r or more of s sequences has order growth log N/(–logλ) where λis the maximal eigenvalue of r-Schur product matrices from among the collections of Markov matrices that generate the sequences. The count Z∗r,s(N, k) of positions that initiate an aligned match of length exceeding k = log N/(–logλ) + x but fail to match at the immediately preceding position has a limiting Poisson distribution. Distributional properties of other long aligned word relationships and patterns are also discussed.

Conditions are given under which a space-time jump Markov process describing the stochastic model of non-linear chemical reactions with diffusion converges to the homogeneous state solution of the corresponding reaction-diffusion equation. The deviation is measured by a central limit theorem. This limit is a distribution-valued Ornstein–Uhlenbeck process and can be represented as the mild solution of a certain stochastic partial differential equation.

Motivated by applications in digital image processing, we discuss information-theoretic bounds to the amount of detail that can be recovered from a defocused, noisy signal. Mathematical models are constructed for test-pattern, defocusing and noise. Using these models, upper bounds are derived for the amount of detail that can be recovered from the degraded signal, using any method of image restoration. The bounds are used to assess the performance of the class of linear restorative procedures. Certain members of the class are shown to be optimal, in the sense that they attain the bounds, while others are shown to be sub-optimal. The effect of smoothness of point-spread function on the amount of resolvable detail is discussed concisely.

A regression approximation of wavelength and amplitude distribution in an almost surely continuous process η (t), is based on a successively more detailed decomposition, η (t) = η n(t) + Δn(t), into one regression term η n on n suitably chosen random quantities, and one residual process Δn. The distances between crossings, maxima, etc., are then approximated by the corresponding quantities in the regression term, and explicit expressions given for the densities of these quantities.

Recently a number of authors have considered general procedures for coupling stochastic systems. If the individual components of a system, when considered in isolation, are found to possess the simplifying feature of either reversibility, quasireversibility or partial balance they can be coupled in such a way that the equilibrium analysis of the system is considerably simpler than one might expect in advance. In particular the system usually exhibits a product-form equilibrium distribution and this is often insensitive to the precise specification of the individual components. It is true, however, that certain kinds of components lose their simplifying feature if the specification of the coupling procedure changes. From a practical point of view it is important, therefore, to determine if, and then under what conditions, the revelant feature is preserved.

In this paper we obtain conditions under which partial balance in a component is preserved and these often amount to the requirement that there exists a quantity which is unaffected by the internal workings of the component in question. We give particular attention to the components of a stratified clustering process as these most often suffer from loss of partial balance.

We model a selection process arising in certain storage problems. A sequence (X1, · ··, Xn) of non-negative, independent and identically distributed random variables is given. F(x) denotes the common distribution of the Xi′s. With F(x) given we seek a decision rule for selecting a maximum number of the Xi′s subject to the following constraints: (1) the sum of the elements selected must not exceed a given constant c > 0, and (2) the Xi′s must be inspected in strict sequence with the decision to accept or reject an element being final at the time it is inspected.

We prove first that there exists such a rule of threshold type, i.e. the ith element inspected is accepted if and only if it is no larger than a threshold which depends only on i and the sum of the elements already accepted. Next, we prove that if F(x) ~ Axα as x → 0 for some A, α> 0, then for fixed c the expected number, En(c), selected by an optimal threshold is characterized by Asymptotics as c → ∞and n → ∞with c/n held fixed are derived, and connections with several closely related, well-known problems are brought out and discussed.

For m infinite-server queues with Poisson input which are connected in a series, a simple proof is given of a formula derived in [3] for the generating function of the joint customer-stationary distribution of the successive numbers of customers a randomly chosen customer finds at his arrival epochs at two queues of the system. In this connection, a shot-noise representation of the queue-length characteristics under consideration is used. Moreover, using this representation, corresonding asymptotic formulas are derived for infinite-server tandem queues with general high-density renewal input.

A closed queueing system has Q service channels and a waiting line. There are Ni customers of type i in the system, i = 1, ···, m, ∑mi=1Ni = N > Q. Q customers are served and Q0 = N – Q stay in the waiting line. Q channels are partitioned into n groups with Qj channels in the jth group, j = 1, ···, n. The service time of the ith type customer by a channel of the jth group is τij ~ Exp (λij). When a customer leaves the channel, it is immediately replaced by another one picked up randomly from the waiting time. The customer which has cleared service joins the waiting line without delay. Let Xij be the number of ith type customers served by jth group channels in equilibrium. An explicit formula for P(Xij = kij, i = 1, ···, m; j = 1, ···, n) is found. It is shown in a form of a local limit theorem that the asymptotic distribution of {Xij} is a multidimensional normal, if Qj/N and Qj/N have positive limits as N → ∞. Formulas for mean values and covariances are given. It turns out that the means of Xij and covariances between Xij and Xrk can be found, using an efficient iterative algorithm, from the deterministic version of the system. A numerical example demonstrates that the normal approximation is rather accurate.

The number of returns to the origin for a simple random walk starting and ending at the origin is reconsidered; closed-form expressions in μ are given for μ3 and μ4.

We derive the Sobel–Uppuluri and Galambos-type extensions of the Bonferroni bounds, and further extensions of the same nature, as consequences of a single non-probabilistic inequality. The methodology follows that of Galambos.

This note provides a useful extension of the Berry–Esseen bound on the error in the normal approximation for shot-noise. The special cases treated are of particular interest in the statistical analysis of Poisson processes and cluster point processes.

A probabilistic proof is given of the fact that the departure process from two initially empty. ·/M/1 queues in tandem is unaffected when the service rates are interchanged. As a consequence of this, we show that when the sum of the service rates at the two queues is held constant the departure process stochastically increases as the service rates become equal. The proofs are based on coupling of reflected random walks.