A discrete Galton–Watson process is modified by an absorbing process, which, within each generation, eliminates a subset of the living particles without leaving offspring. The absorbing process will be only roughly specified by its expected efficiency on the associated process. We give a sharp sufficient condition for the joint process to be extinguished with probability one and give, after an easy generalization, an example for possible biomedical applications.

This paper deals with the preservation of strong forms of mixing of point processes under the clustering operation. Strong, ϕ -, and *-mixing possess an increasing degree of uniformity of their asymptotic independence, and this turns out to be a significant factor in their preservation. In particular, it is indicated that ϕ-mixing may be maintained only under very stringent conditions (bounded clusters), whereas strong mixing is maintained under considerably milder conditions.

In this paper we use the properties of the covariance matrix of a Gaussian Markovian family to give a probabilistic proof of a theorem about inverses of tridiagonal matrices.

A draining is a stochastic process defined by an urn scheme where the successive drawings are made without replacement and according to a drawing algorithm associated with a weighted graph. A draining is said to be a strong one if the number of balls which can be drawn from the urn (under the algorithm) converges in probability to the total number of balls in the urn at the beginning of the drawing when it goes to infinity. In particular, drainings associated with complete graphs having equal weights are found to be strong and for some others associated weighted graphs, strong drainings exist.

Fix a bivariate distribution F on X × Y, considered as a pair (α, {Fx}), where α is a marginal distribution on X and {Fx} is a collection of conditional distributions on Y. For essentially every (β,{Gx}) satisfying a certain pair of moment conditions determined by (α, {Fx}), J(β, {Fx}) ≦ J(α, {Fx}) ≦ J(α, {Gx}), where J is mutual information. This relates to two sorts of extremizations of mutual information of relevance to communication theory and statistics.

Let 0 ≦ T1 ≦ T2 ≦ ·· · represent the epochs in time of occurrences of events of a point process N(t) with N(t) = sup{k : Tk ≦ t}, t ≧ 0. Besides certain mild conditions on the process N(t) (see Conditions (A1)– (A3) in the text) we assume that for every k ≧ 1, as t →∞, the vector (t – TN(t), t – TN(t)–1, · ··, t – TN(t)–k+1) converges in law to a k-dimensional distribution which coincides with that of a random vector ξ k = (ξ1, · ··, ξ k) necessarily satisfying P(0 ≦ ξ1 ≦ ξ2 ≦ ·· ·≦ ξk) = 1. Let R(t) be an arbitrary function defined for t ≧ 0, satisfying 0 ≦ R(t) ≦ 1, ∀0 ≦ t <∞, and certain mild conditions (see Conditions (B1)– (B4) in the text). Then among other results, it is shown that

The paper also deals with conditions under which the limit (∗) will be positive. The results are applied to several point processes and to the situations where the role of R(t) is taken over by an appropriate transform such as a probability generating function, where conditions are given under which the limit (∗) itself will be a transform of an honest distribution. Finally the results are applied to the study of certain characteristics of the GI/G/∞ queue apparently not studied before.

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).

A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

Let {ξk : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ1} = μ ≠ 0. Form the random walk {Sn : n ≧ 0} by setting S0, Sn = ξ1 + ξ2 + ··· + ξn, n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ1) that the finite-dimensional distributions of Xn, conditioned on n < N < ∞ converge to those of the Brownian excursion process.

This paper is concerned with the characterization of the cumulative pensionable service over an individual's working life that is made up of random lengths of service in different employments in a given industry, under partial coverage, transferability, and a uniform vesting rule. This characterization uses some results that are developed in the paper involving a functional and cumulative constrained sojourn times (constrained in the sense that if a sojourn time is less than a given constant it is not counted) in semi-Markov processes.

We use a result on integration by parts for stochastic integrals together with a technique developed by Getoor in [6], to express entrance—exit distributions for a standard process X, and a set Φ which is the support of a continuous additive functional C, in terms of the infinitesimal generators of semigroups associated with the time-changed process (Xτt), where (τt) is the right-continuous inverse of C.

In a recent paper Green (1976) obtained some conditional limit theorems for the absorption time of left-continuous random walk. His methods require that in the driftless case the increment distribution has exponentially decreasing tails and that the same is true for a transformed distribution in the case of negative drift.

Here we take a different approach which will produce Green's results under minimal conditions. Limit theorems are given for the maximum as the initial position of the random walk tends to infinity.

Wold's point process is briefly introduced and its forward equation is derived in terms of an integro-differential equation which is used to obtain the ‘renewal' function. An example of this family of processes is given in which each interval is exponentially distributed. The theory of diagonal expansion for a bivariate distribution is used to obtain the numerical estimate of the spectrum of counts.

Frequency domain estimation and tests of hypotheses are considered for a general multivariate distributed lag regression model. The usual vector of regression functions is replaced by a matrix of almost periodic functions, a case for which the terms appearing in the frequency domain estimators have well-defined limits. Asymptotic distributions and consistency are established for the regression coefficients and error spectra. A test statistic is proposed for the no regression hypothesis which is asymptotically central under the null hypothesis and converges to infinity under the alternative. In the multivariate case, a product of beta-distributed variables is obtained.

An ‘oscillating' version of Brownian motion is defined and studied. ‘Ordinary' Brownian motion and ‘reflecting' Brownian motion are shown to arise as special cases. Transition densities, first-passage time distributions, and occupation time distributions for the process are obtained explicitly. Convergence of a simple oscillating random walk to an oscillating Brownian motion process is established by using results of Stone (1963).

For finite Markov chains, the concepts of ergodicity and strong ergodicity are equivalent, but this is not necessarily the case when the state space is infinite. In this note we give some new characterizations of strong ergodicity. These lead to simple necessary or sufficient criteria for strong ergodicity, which readily enable us to classify a number of examples.

We define a vector-valued scheduled maxima sequence M by considering simultaneously the maxima of several i.i.d. sequences, with the number of observations considered from each sequence at any time determined by a random scheduling sequence J. It is shown that the max-min (vector) sequence derived from i.i.d. can be represented as a mixture of scheduled maxima sequences, giving results for this sequence and the range A functional limit theorem for the scheduled maxima sequence shows convergence to independent extremal processes. Embedding in a scheduled extremal process gives strong laws, central limit theorems, and laws of the iterated logarithm for the record time of the scheduled maxima sequence, and hence for the max-min sequence and the range.

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

Place n arcs of equal lengths randomly on the circumference of a circle, and let C denote the proportion covered. The moments of C (moments of coverage) are found by solving a recursive integral equation, and a formula is derived for the cumulative distribution function. The asymptotic distribution of C for large n is explored, and is shown to be related to the exponential distribution.

A study is made of the extremal process generated by i.i.d. random variables appearing at the events of a non-homogeneous Poisson process, 𝒫. If 𝒫 has an exponentially increasing rate function, then records eventually occur in a homogeneous Poisson process. The size of the latest record has a classical extreme value distribution.

Let {Nt}t >0 be a renewal counting process (cf. Parzen (1962), p. 160) with underlying failure times let be a sequence of non-negative random variables and {Zt}t >0 an associated cumulative process, i.e. if Nt = 1, 2, …, and Zt = 0, if Nt = 0. By convention set Z0 = 0. Consider the maximum increment of the process {Zt}t >0 in [0, T] over a time K, 0 < K < T, divided by K. Under appropriate conditions it is shown that for a wide range of numbers a there exist constants C(a), uniquely determined by a and the distributions of the Xi's and Yj's, such that D(T, C log T) converges to a with probability 1. This result provides a renewal theoretic variant of Erdös and Rényi's (1970) ‘new law of large numbers’.