Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

In this paper exchangeable bisexual models with fixed population size and non-overlapping generations are introduced. Each generation consists of N pairs of individuals. The pairs of a generation have altogether 2N children. These individuals form randomly the N pairs of the next generation. The extinction probability of the descendants of a fixed number of pairs of generation 0 is discussed. Under suitable conditions it can be approximately described by the extinction probability of a Galton–Watson process, if the population size is large. Special examples are a bisexual Wright–Fisher model and models with a uniformly bounded number of children of a pair.

The stationary version of the infinitely many neutral alleles diffusion model, as studied by Ethier and Kurtz, is constructed and analyzed using the theory of Dirichlet forms. We prove that the ‘boundary' of the state space is hit by the process iff θ < 1, where θ is the mutation parameter.

This paper considers a carrier-borne epidemic in continuous time with m + 1 > 2 stages of infection. The carriers U(t) follow a pure death process, mixing homogeneously with susceptibles X0(t), and infectives Xi(t) in stages 1≦i≦m of infection. The infectives progress through consecutive stages of infection after each contact with the carriers. It is shown that under certain conditions {X0(t), X1(t), · ··, Xm(t) U(t); t≧0} is an (m + 2)-variate Markov chain, and the partial differential equation for its probability generating function derived. This can be solved after a transfomation of variables, and the probability of survivors at the end of the epidemic found.

Quasi-stationary distributions are considered in their own right, and from the standpoint of finite approximations, for absorbing birth-death processes. Results on convergence of finite quasi-stationary distributions and a stochastic bound for an infinite quasi-stationary distribution are obtained. These results are akin to those of Keilson and Ramaswamy (1984). The methodology is a synthesis of Good (1968) and Cavender (1978).

A probabilistic predator-prey model is constructed using linked discrete-time branching-type processes. A necessary and sufficient condition for positive probability of survival of both populations is given.

We consider the limit distribution of maxima and point processes, connected with crossings of an increasing level, for a sequence of Gaussian stationary processes. As an application we investigate the limit distribution of the error of approximation of Gaussian stationary periodic processes by random trigonometric polynomials in the uniform metric.

Invariant stochastic properties are studied for circuit processes, i.e. processes whose sample paths can be represented by a denumerable set of overlapping directed circuits c in a denumerable set S and a set of positive weights , when the family of the weights varies.

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations.

The points of a graph G will form clusters as a result of a flow process. Initially, points i of G own resources xi which are i.i.d. random real numbers. Afterwards, resources flow between points, but always from a point to a neighbor that has accumulated a larger total resource. Thus points with small resource tend to lose it and points with large resource tend to gain. Eventually the flow stops with only two kinds of points, nulls with no resource left and absorbers with such large resource that no neighbor can take it. The final resource at an absorber is a sum of certain initial resources xi, and the corresponding points i form one cluster. Analytical results are obtainable when G is the chain of integer points on the line. Probability distributions are derived for the distance between consecutive absorbers and the size of a cluster. Indeed these distributions do not involve the given distribution for the xi. The Laplace transform of the distribution of final resources at absorbers is derived but the distribution itself is obtained by a simulation. For general graphs G only partial results are obtained.

Consider a time-homogeneous Markov branching process. We construct reduced processes, based on whether the length of line of descent of particles of this process are (a) greater than or (b) at most equal to, τ units of time, for some fixed τ ≧ 0. We show that in both cases the reduced processes retain the branching property, but the latter does not retain the time homogeneity. We investigate finite-time and asymptotic properties of the reduced processes. Based on a realization of the original process and a realization of a reduced process, observed continuously over a time interval [0, T] for T > 0, we propose estimators for the different parameters involved, including qτ, the probability that the original process becomes extinct before τ units of time, and f(j)(qτ), the jth derivative of the offspring probability generating function f(s) at qτ when qτ is known. We study the properties of these estimators and derive their asymptotic distributions, under the assumption that the original process is supercritical.

The joint limiting distribution of suitably normalized partial sums and maxima in a stationary strong mixing sequence with finite variance is derived. It is found that in the limit the two components are independent. This generalizes Chow and Teugels' result for independent sequences. Motivation for the present study comes from a statistical problem in the analysis of extreme winds.

If Xi, i = 1, ···, n are independent exponential random variables with parameters λ1, · ··, λ n, and if Yi, i = 1, ···, n are independent exponential random variables with common parameter equal to (λ1 + · ·· + λ n)/n, then there is a monotone coupling of the order statistics X(1), · ··, X(n) and Y(1), · ··, Y(n); that is, it is possible to construct on a common probability space random variables X′i, Y′i, i = 1, ···, n, such that for each i, Y′(i)≦X′(i) a.s., where the law of the X′i (respectively, the Y′i) is the same as the law of the Xi (respectively, the Yi.) This result is due to Proschan and Sethuraman, and independently to Ball. We shall here prove an extension to a more general class of distributions for which the failure rate function r(x) is decreasing, and xr(x) is increasing. This very strong order relation allows comparison of properties of epidemic processes where rates of infection are not uniform with the corresponding properties for the homogeneous case. We further prove that for a sequence Zi, i = 1, ···, n of independent random variables whose failure rates at any time add to 1, the order statistics are stochastically larger than the order statistics of a sample of n independent exponential random variables of mean n, but that the strong monotone coupling referred to above is impossible in general.

It is shown that the stationary, autoregressive, Markovian minification processes introduced by Tavares and Sim can be extended to give processes with marginal distributions other than the exponential and Weibull distributions. Necessary and sufficient conditions on the hazard rate of the marginal distributions are given for a minification process to exist. Results are given for the derivation of the autocorrelation function; these correct the expression for the Weibull given by Sim. Monotonic transformations of the minification processes are also discussed and generate a whole new class of autoregressive processes with fixed marginal distributions. Stationary processes generated by a maximum operation are also introduced and a comparison of three different Markovian processes with uniform marginal distributions is given.

Points arrive in succession on an interval and immediately ‘cover' a region of length ½ to each side (less if they are close to the boundary or to a covered part). The location of a new point is uniformly distributed on the uncovered parts. We study the mean and variance of the total number of points ever formed, in particular as a → 0, in which case we also establish asymptotic normality.

Sufficient conditions are given under which two simple point processes on the positive half-line can be stochastically compared as random elements of D(0,∞) or R∞+ Using a martingale approach to point processes, the conditions are proposed via a compensator function family. Appropriate versions of the processes being compared are constructed on the same probability space. The results are illustrated by replacement policies and semi-Markov point processes.

Let {Xk} be an i.i.d. sequence taking values in ℝ2 with the radial and spherical components independent and the radial component having a distribution with slowly varying tail. The number of extreme points in the convex hull of {X1, · ··, Xn} is shown to have a limiting distribution which is obtained explicitly. Precise information about the mean and variance of the limit distribution is obtained.