A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector process X(t) = (X1(t), · ··, XN(t)) in which the arrival, departure and transfer rates for the states {X(t) = n} may be written in the form α i(ni), βi(ni) and γ ij(ni, nj), respectively, where n = (n1, · ··, nN). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.

We provide a unified probabilistic approach to the distribution of total size and total area under the trajectory of infectives for a general stochastic epidemic with any specified distribution of the infectious period. The key tool is a Wald&s identity for the epidemic process. The generalisation of our results to epidemics spreading amongst a heterogeneous population is straightforward.

Consider the sequence of partial sums of a sequence of i.i.d. random variables with positive expectation.

We study various random quantities defined by the sequence of partial sums, e.g. the time at which the first or last crossing of a given level occurs, the value of the partial sum immediately before or after the crossing, the minimum of all partial sums. Necessary and sufficient conditions are given for the existence of moments of these quantities.

A new coalescent is introduced to study the genealogy of a sample from the infinite-alleles model of population genetics. This coalescent also records the age ordering of alleles in the sample. The distribution of this process is found explicitly for the Moran model, and is shown to be robust for a wide class of reproductive schemes.

Properties of the ages themselves and the relationship between ages and class sizes then follow readily.

For all uniformly bounded sequences of independent random variables X1, X2, ···, a complete comparison is made between the optimal value V(X1, X2, ···) = sup {EXt:t is an (a.e.) finite stop rule for X1,X2, ···} and , where Mi(X1,X2, ···) is the ith largest order statistic for X1, X2, ··· In particular, for k> 1, the set of ordered pairs {(x, y):x = V(X1, X2, ···) and for some independent random variables X1, X2, ··· taking values in [0, 1]} is precisely the set , where Bk(0) = 0, Bk(1) = 1, and for The result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the game V(X1, X2, ···) and the prophet&s constrained maxima expectation of the game . Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

Batches of immigrants arrive in a region at event times of a renewal process and individuals grow according to a Bellman-Harris branching process. Tribal emigration allows the possibility that all descendants of a group of immigrants collectively leave the region at some instant.

A number of results are derived giving conditions for the existence of a limiting distribution for the population size. These conditions can be given either in terms of the immigration distribution or in terms of the distribution of emigration times. Some limit theorems are obtained when the latter conditions are not fulfilled.

We provide a framework for interconnecting a collection of reversible Markov processes in such a way that the resulting process has a product-form invariant measure with respect to which the process is reversible. A number of examples are discussed including Kingman&s reversible migration process, interconnected random walks and stratified clustering processes.

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

We study the sample path behaviour of χ2 processes in the neighbourhood of their level crossings and extrema via the development of Slepian model processes. The results, aside from being of particular interest in the study of χ2 processes, have a general interest insofar as they indicate which properties of Gaussian processes (which have been heavily researched in this regard) are mirrored or lost when the assumption of normality is not made. We place particular emphasis on the behaviour of χ2 processes at both high and low levels, these being of considerable practical importance. We also extend previous results on the asymptotic Poisson form of the point process of high maxima to include also low minima (which are in a different domain of attraction) thus closing a gap in the theory of χ2 processes.

A method is reviewed for proving weak convergence in a function-space setting when regular variation is a sufficient condition. Point processes and weak convergence techniques involving continuity arguments play a central role. The method is dimensionless and holds computations to a minimum. Many applications of the methods to processes derived from sums and maxima are given.

Spatial point processes are considered whose points are subjected to certain classes of affine transformations indexed by some variable, T. Under some hypotheses, for large T integrals with respect to such a point process behave approximately as if the process were Poisson. Under stronger hypotheses, the transformed process converges as a process to a Poisson process. The result gives the asymptotic distribution of certain density estimates.

Optimal causal policies maximizing the time-average reward over a semi-Markov decision process (SMDP), subject to a hard constraint on a time-average cost, are considered. Rewards and costs depend on the state and action, and contain running as well as switching components. It is supposed that the state space of the SMDP is finite, and the action space compact metric. The policy determines an action at each transition point of the SMDP.

Under an accessibility hypothesis, several notions of time average are equivalent. A Lagrange multiplier formulation involving a dynamic programming equation is utilized to relate the constrained optimization to an unconstrained optimization parametrized by the multiplier. This approach leads to a proof for the existence of a semi-simple optimal constrained policy. That is, there is at most one state for which the action is randomized between two possibilities; at all other states, an action is uniquely chosen for each state. Affine forms for the rewards, costs and transition probabilities further reduce the optimal constrained policy to ‘almost bang-bang’ form, in which the optimal policy is not randomized, and is bang-bang except perhaps at one state. Under the same assumptions, one can alternatively find an optimal constrained policy that is strictly bang-bang, but may be randomized at one state. Application is made to flow control of a birth-and-death process (e.g., an M/M/s queue); under certain monotonicity restrictions on the reward and cost structure the preceding results apply, and in addition there is a simple acceptance region.

We study a rule of growing a sequence {tn} of finite subtrees of an infinite m-ary tree T. Independent copies {ω (n)} of a Bernoulli-type process ω on m letters are used to trace out a sequence of paths in T. The tree tn is obtained by cutting each , at the first node such that at most σ paths out of , pass through it. Denote by Hn the length of the longest path, hn the length of the shortest path, and Ln the length of the randomly chosen path in tn. It is shown that, in probability, Hn – logan = O(1), hn – logb (n/log n) = 0(1), (or hn – logb (n/log log n) = O(1)), and that is asymptotically normal. The parameters a, b, c depend on the distribution of ω and, in case of a, also on σ. These estimates describe respectively the worst, the best and the typical case behavior of a ‘trie’ search algorithm for a dictionary-type information retrieval system, with σ being the capacity of a page.

The Akaike information criterion, AIC, for autoregressive model selection is derived by adopting −2T times the expected predictive density of a future observation of an independent process as a loss function, where T is the length of the observed time series. The conditions under which AIC provides an asymptotically unbiased estimator of the corresponding risk function are derived. When the unbiasedness property fails, the use of AIC is justified heuristically. However, a method for estimating the risk function, which is applicable for all fitted orders, is given. A derivation of the generalized information criterion, AICα, is also given; the loss function used being obtained by a modification of the Kullback-Leibler information measure. Results paralleling those for AIC are also obtained for the AICα criterion.

The problem of identifying solutions of general convolution equations relative to a group has been studied in two classical papers by Choquet and Deny (1960) and Deny (1961). Recently, Lau and Rao (1982) have considered the analogous problem relative to a certain semigroup of the real line, which extends the results of Marsaglia and Tubilla (1975) and a lemma of Shanbhag (1977). The extended versions of Deny&s theorem contained in the papers by Lau and Rao, and Shanbhag (which we refer to as LRS theorems) yield as special cases improved versions of several characterizations of exponential, Weibull, stable, Pareto, geometric, Poisson and negative binomial distributions obtained by various authors during the last few years. In this paper we review some of the recent contributions to characterization of probability distributions (whose authors do not seem to be aware of LRS theorems or special cases existing earlier) and show how improved versions of these results follow as immediate corollaries to LRS theorems. We also give a short proof of Lau–Rao theorem based on Deny&s theorem and thus establish a direct link between the results of Deny (1961) and those of Lau and Rao (1982). A variant of Lau–Rao theorem is proved and applied to some characterization problems.

Fortunately traditional reliability theory, where the system and the components are always described simply as functioning or failed, is on the way to being replaced by a theory for multistate systems of multistate components. However, there is a need for several convincing case studies demonstrating the practicability of the generalizations introduced. In this paper an electrical power generation system for two nearby oilrigs will be discussed. The amounts of power that may possibly be supplied to the two oilrigs are considered as system states.

Most work in the theory of dams has dealt with deterministic withdrawals. Some more recent work, begun by Moran (1969), has been devoted to conditionally deterministic withdrawals which depend on the content of the dam, or in some cases on the last input. We consider in this paper an infinite dam fed by discrete additive inputs, under a totally random withdrawal policy. Explicit expressions are obtained for the time-dependent content distributions. Probabilities of emptiness are computed and stationary results are derived.

Some simple models are described which may be used for the modelling or generation of sequences of dependent discrete random variates with negative binomial and geometric univariate marginal distributions. The models are developed as analogues of well-known continuous variate models for gamma and negative exponential variates. The analogy arises naturally from a consideration of self-decomposability for discrete random variables. An alternative derivation is also given wherein both the continuous and the discrete variate processes arise simultaneously as measures on a process of overlapping intervals. The former is the process of interval lengths and the latter is a process of counts on these intervals.

Let X1, ···, Xn be i.i.d. random variables defined in ℝd having common continuous density f(x), and let Rij be the rank of Xj in the ordered list of distances from X¡. Both the mutual neighbor probabilities p1(r, s) = P(R12 = r, R21 = s) and the neighbor-sharing probabilities p2(r, s) = P(R13 = r, R23 = s) are studied from an asymptotic viewpoint. Infinite-dimensional limits are found for both situations and take particularly simple forms. Both cases exhibit considerable stability across dimensions and thus are well approximated by their infinite-dimensional values. Tables are provided to support the results given.

A triangle with vertices z1, z2, z3 in the complex plane may be denoted by a vector Z, Z = [z1, z2, z3]t. From a sequence of independent and identically distributed 3×3 circulants {Cj}∞1, we may generate from Z1 the sequence of vectors or triangles {Zj}∞1, by the rule Zj = CjZj–1 (j> 1), Z1=Z. The ‘shape’ of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence {Zj}∞1. In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles.