In this paper we consider an independent and identically distributed sequence {Yn} with common distribution function F(x) and a random variable X0, independent of the Yi's, and define a Markovian sequence {Xn} as Xi = X0, if i = 0, Xi = k max{Xi− 1, Yi}, if i ≧ 1, k ∈ R, 0 < k < 1. For this sequence we evaluate basic distributional formulas and give conditions on F(x) for the sequence to possess a stationary distribution. We prove that for any distribution function H(x) with left endpoint greater than or equal to zero for which log H(ex) is concave it is possible to construct such a stationary sequence with marginal distributions equal to it. We study the limit laws for extremes and kth order statistics.

We obtain an approximation to the mean time to extinction and to the quasi-stationary distribution for the standard S–I–S epidemic model introduced by Weiss and Dishon (1971). These results are a combination and extension of the results of Norden (1982) for the stochastic logistic model, Oppenheim et al. (1977) for a model on chemical reactions, Cavender (1978) for the birth-and-death processes and Bartholomew (1976) for social diffusion processes.

We consider a multitype population-size-dependent branching process in discrete time. A process is considered to be near-critical if the mean matrices of offspring distributions approach the mean matrix of a critical process as the population size increases. We show that if the second moments of offspring distributions stabilize as the population size increases, and the limiting variances are not too large in comparison with the deviation of the means from criticality, then the extinction probability is less than 1 and the process grows arithmetically fast, in the sense that any linear combination which is not orthogonal to the left eigenvector of the limiting mean matrix grows linearly to a limit distribution. We identify cases when the limiting distribution is gamma. A result on transience of multidimensional Markov chains is also given.

In Gadag and Rajarshi (1987), we studied a bivariate (multitype) branching process based on infinite and finite lines of descent, of particles of a supercritical one-dimensional (multitype) Galton-Watson branching process (GWBP). In this paper, we discuss a few more meaningful and interesting univariate and multitype branching processes, based on exact progeny lengths of particles in a GWBP. Our constructions relax the assumption of supercriticality made in Gadag and Rajarshi (1987). We investigate some finite-time and asymptotic results of these processes in some details and relate them to the original process. These results are then used to propose new and better estimates of the offspring mean. An illustration based on the branching process of the white male population of the USA is also given. We believe that our work offers a rather finer understanding of the branching property.

Consider a population of distinct species Sj, j∈J, members of which are selected at different time points T1, T2,· ··, one at each time. Assume linear costs per unit of time and that a reward is earned at each discovery epoch of a new species. We treat the problem of finding a selection rule which maximizes the expected payoff. As the times between successive selections are supposed to be continuous random variables, we are dealing with a continuous-time optimal stopping problem which is the natural generalization of the one Rasmussen and Starr (1979) have investigated; namely, the corresponding problem with fixed times between successive selections. However, in contrast to their discrete-time setting the derivation of an optimal strategy appears to be much harder in our model as generally we are no longer in the monotone case.

This note gives a general point process formulation for this problem, leading in particular to an equivalent stopping problem via stochastic intensities which is easier to handle. Then we present a formal derivation of the optimal stopping time under the stronger assumption of i.i.d. (X1, A1) (X2, A2), · ·· where Xn gives the label (j for Sj) of the species selected at Tn and An denotes the time between the nth and (n – 1)th selection, i.e. An = Tn – Tn–1. In the case where even Xn and An are independent and An has an IFR (increasing failure rate) distribution, an explicit solution for the optimal strategy is derived as a simple consequence.

We analyse the conditions under which the aggregated process constructed from an homogeneous Markov chain over a given partition of the state space is also Markov homogeneous. The past work on the subject is revised and new properties are obtained.

Lower and upper bounds of degree m for the probability of the union of n not necessarily exchangeable events are established. These bounds may be constructed to improve the Bonferroni and the Sobel–Uppuluri bounds.

An application to equi-correlated multivariate normal distribution is given.

Large stochastic systems of marked particles are considered. These ‘populations' grow according to pairwise mutually catalytic reactions and in addition particles may exchange their type (mark) by a sampling procedure. We are interested in the explosive behavior of the model and its high density properties (law of large numbers).

Let ∊1∊2, · ··, be a stationary sequence satisfying the weak long-range dependence condition Δ (un(τ)) of [3] for every τ > 0, where nP(∊1> un(τ))→ τ. Assume only that P (there are j exceedances of un(τ) by ∊1, ∊2, · ··, ∊n) converges for all j with 0≦j≦υ<∞ and a given fixedτ. Then the same holds for every τ> 0. For 0≦j≦υ the limit is P(X = j) where X is compound Poisson and the multiplicity distribution is independent ofτ. These results are extended to more general levels un and to cases where the joint distribution of the numbers of exceedances of several levels is considered. The limiting distributions of linearly normalized extreme order statistics are derived as a corollary. An application to insurance claim data is discussed.

Special symmetry conditions on the transition p.d.f. of one-dimensional time-homogeneous diffusion processes with natural boundaries are investigated and exploited to derive closed-form results concerning the transition p.d.f.'s in the presence of absorbing and reflecting boundaries and the first-passage-time p.d.f. through time-dependent boundaries.

The theory of insensitivity within generalised semi-Markov processes is extended to cover classes of models in which the generally distributed lifetimes can be terminated prematurely by the deaths of negative exponentially distributed lifetimes. As a consequence of this approach it is shown that there exist classes of processes which are insensitive with respect to characteristics of the general distributions other than the mean. Two examples are given. The first is an analysis of networks of queues in which the generally distributed service times can be interrupted with resulting changes in routing probabilities. The second is a model for the effect of disturbances on the evolution of a vegetation community.

The constructive solution to the problem of representing a strictly stationary Markov chain ζ with a countable infinity of r-sequences (i1, i2, · ··, ir), r > 1, as states by a class of directed weighted circuits is given. Associating the chain ζ with its dual chain η having reversed states and the same transition law, a connection with physical laws that govern diffusion of electrical current through a directed planar network with r-series-connected nodes is shown.

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.

A Slepian model for the local behaviour near the level upcrossings of a x2-process with dependent Gaussian components is presented. In case of independent components, this model is shown to take on a rather simple form, thereby simplifying earlier results by Aronowich and Adler.

The Slepian model is applied to the envelope of a stationary Gaussian process and used to approximate the probability of ‘empty' envelope upcrossings, i.e. the probability that an envelope upcrossing is not followed by a level crossing in the original process.

The random splitting model of Lloyd and Williams (1988) is generalised, to allow first beta splitting distributions and secondly, discrete splitting distributions. Analogues of some earlier results are developed. In particular, the equality of the average value of the largest split and the probability that this is achieved at the first split continues to hold.

We consider a class of secretary problems in which the order of arrival of candidates is no longer uniformly distributed. By a suitable embedding in a time-transformed extremal process it is shown that the asymptotic winning probability is again 1/e as in the classical situation. Extensions of the problem to more than one choice are also considered.

The problem of the optimal control of the Wiener process in Rn is considered. The optimal value of the control is obtained from the mathematical expectation of a quantity defined in terms of the moment and the place where the uncontrolled process hits the boundary of the continuation region for the first time. Explicit results are presented.

A non-degenerate distribution function F is called maximum stable with random sample size if there exist positive integer random variables Nn, n = 1, 2, ···, with P(Nn = 1) less than 1 and tending to 1 as n → ∞ and such that F and the distribution function of the maximum value of Nn independent observations from F (and independent of Nn) are of the same type for every index n. By proving the converse of an earlier result of the author, it is shown that the set of all maximum stable distribution functions with random sample size consists of all distribution functions F satisfying where c2, c3, · ·· are arbitrary non-negative constants with 0 < c2 + c3 + · ·· <∞, and all distribution functions G and H defined by F(x)= G(c + exp(x)) and F(x) = H(c – exp(–x)), –∞ < x <∞, where c is an arbitrary real constant.

This paper gives the necessary and sufficient condition for the second-order stationarity of lower triangular bilinear models; this condition is in terms of norms of higher-order transfer functions. The spectral density function of the process is evaluated in terms of the transfer functions, and it is shown that the second-order structure is similar to a linear ARMA model with uncorrelated errors. The best linear predictor is obtained and it is shown that the variance of the prediction error is always greater than the optimal prediction error variance obtained from the bilinear model. An expression for the bispectral density function is also obtained.

We verify and extend a conjecture of Purdue (1981) concerning the stochastic monotonicity of absorption times in a class of compartmental urn models. We also describe the effect of increased variability in the reinforcement sizes. Finally, we investigate variability in the content process for large populations. In many applications, compartmental models substantially under-represent the variability observed in the data, so that there has been considerable interest in modifying the model to increase the variability. We show that the squared coefficient of variation of the content is not asymptotically negligible when both the size and the variability of the reinforcements are of the same order as the initial population.