We introduce a new permutation statistic, which we call the k-window statistic, for permutations of $\{1,2,\ldots, n\}$. The aim is to add to the repertoire of potentially useful tools in the design of efficient algorithms. The k-window statistic counts the number of elements that are at a displacement at most k from their natural positions in the ascendingly sorted permutation. We find the exact mean and exact variance of the k-window statistic, revealing phases in the mean and variance in the different regimes of k. We find a weak (or even strong) law of large numbers in the regime where $k=k_n$ is growing but remains ${\mathrm{o}} (n)$. Via a representation of the statistic into dependent heterogeneous weakly correlated indicators, we establish the complete spectrum of distributions as $k_n$ spans its entire range using an appropriate method for each regime.

We consider a generalized form of the coupon collection problem in which a random number, S, of balls is drawn at each stage from an urn initially containing n white balls (coupons). Each white ball drawn is colored red and returned to the urn; red balls drawn are simply returned to the urn. The question considered is then: how many white balls (uncollected coupons) remain in the urn after the k n draws? Our analysis is asymptotic as n → ∞. We concentrate on the case when k n draws are made, where k n / n → ∞ (the superlinear case), although we sketch known results for other ranges of k n . A Gaussian limit is obtained via a martingale representation for the lower superlinear range, and a Poisson limit is derived for the upper boundary of this range via the Chen-Stein approximation.

A sequence of independent Bernoulli random variables with success probabilities a / (a + b + k − 1), k = 1, 2, 3, …, is embedded in a marked Poisson process with intensity 1. Using this, conditional Poisson limits follow for counts of failure strings.

The limiting distributions of the sums of the lengths of four different kinds of runs of consecutive successes of Markov Bernoulli sequences are derived. It is shown that the limits are convolutions of several distributions involving the Bernoulli, the geometric and the Poisson or compound Poisson distributions.

Let Xt , be a Markov random field assuming values in RM. Let In be a rectangular box in Zm with its center at 0 and corner points with coordinates ±n. Let (An ) be a sequence of measurable subsets of RM such that neighborhood of t) → 0, for n → ∞; and let fn (x) be the indicator of An. Under appropriate conditions on the nearest-neighbor distributions of (Xt ), the conditional distribution of given the values of Xs , for s on the boundary of In , converges to the Poisson distribution. An immediate application is an extreme value limit theorem for a real-valued Markov random field.

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.

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.

Poisson limit theorems for U-statistics are studied. A general rate of convergence is obtained; this rate is improved for the special case where the U-statistic arises from the consideration of distances between uniformly distributed points in a well-behaved plane region.