Lars Holst

On the distribution of search cost for the move-to-front rule

A file of records, each with an associated request probability, is dynamically maintained as a serial list. Successive requests are mutually independent. The list is reordered according to the move-to-front (MTF) rule: The requested record is moved to the front of the list. We derive the stationary distribution of search cost (=depth of requested item) by embedding in Poisson processes and derive certain finite-time stochastic ordering results for the MTF chain so embedded. A connection with cache fault probabilities is discussed. We also establish a Schur-concavity result for stationary expected search cost.

Some applications of the Stein-Chen method for proving Poisson convergence

Let W be a sum of Bernoulli random variables and U, a Poisson random variable having the same mean A = EW. Using the Stein-Chen method and suitable couplings, general upper bounds for the variational distance between W and U; are given. These bounds are applied to problems of occupancy, using sampling with and without replacement and P61ya sampling, of capture-recapture, of spacings and of matching and m6nage.

A UNIFIED APPROACH TO LIMIT THEOREMS FOR URN MODELS

An urn contains A balls of each of N colours. At random n balls are drawn in succession without replacement, with replacement or with replacement together with S new balls of the same colour. Let X, be the number of drawn balls having colour k, k = 1, - - -, N. For a given function f the characteristic function of the random variable Z, = f(X)+ - -... + f(X,), M N, is derived. A limit theorem for ZM when M, N, n - oo is proved by a general method. The theorem covers many special cases discussed separately in the literature. As applications of the theorem limit distributions are obtained for some occupancy problems and for dispersion statistics for the binomial, Poisson and negative-binomial distribution.

Asymptotic spacings theory with applications to the two-sample problem

The asymptotic distribution theory of test statistics which are functions of spacings is studied here. Distribution theory under appropriate close alternatives is also derived and used to find the locally most powerful spacing tests. For the two-sample problem, which is to test if two independent samples are from the same population, test statistics which are based on “spacing-frequencies” (i.e., the numbers of observations of one sample which fall in between the spacings made by the other sample) are utilized. The general asymptotic distribution theory of such statistics is studied both under the null hypothesis and under a sequence of close alternatives.

The general birthday problem

The general birthday problem with unlike birth probabilities and the waiting time N until c people with the same birthday have been obtained is studied in this article. It is shown that N is stochastically largest when the birth probabilities are equal. By embedding in Poisson processes it is shown how the moments of N can be expressed in moments of the minimum of gamma random variables.

ON MULTIPLE COVERING OF A CIRCLE WITH RANDOM ARCS

On a circle of unit circumference arcs of length a are placed at random. Let N,, be equal to the necessary number of arcs to cover at least the length 1 - p, 0 _ p < 1, of the circumference at least m ( = 1) times. In the present paper limit distributions of N,, are derived when a --0. Some results for spacings are also obtained.

EXTREME VALUE DISTRIBUTIONS FOR RANDOM COUPON COLLECTOR AND BIRTHDAY PROBLEMS

Take n independent copies of a strictly positive random variable X and divide each copy with the sum of the copies, thus obtaining n random probabilities summing to one. These probabilities are used in independent multinomial trials with n outcomes. Let Nn(N*n) be the number of trials needed until each (some) outcome has occurred at least c times. By embedding the sampling procedure in a Poisson point process the distributions of Nn and N*n can be expressed using extremes of independent identically distributed random variables. Using this, asymptotic distributions as n → ∞ are obtained from classical extreme value theory. The limits are determined by the behavior of the Laplace transform of X close to the origin or at infinity. Some examples are studied in detail.

A limit theorem for the replication time of a DNA molecule

The DNA of higher animals replicates by an interesting mechanism. Enzymes recognise specific sites randomly scattered on the molecule and establish a bidirectional process of unwinding and replication from these sites. We investigate the limiting distribution of the completion time for this process by considering related coverage problems investigated by Janson (1983) and Hall (1988).

ON THE RANDOM COVERAGE OF THE CIRCLE

Place n arcs of equal length a uniformly at random on the circumference of a circle. We discuss the joint limit distributions of the number of gaps, the uncovered proportion of the circle and the lengths of the largest gap and of the smallest gap, depending on how a --0 as n -+ oo. We show that the results may be proved in a unified and simple way by using a result of Le Cam.

On Numbers Related to Partitions of Unlike Objects and Occupancy Problems

Consider n unlike objects and sets of positive integers A and B. Let S(n, A, B) be the number of partitions into j classes of sizes in the set B and j ∈ A. By simple probabilistic arguments the exponential generating function for {S(n, A, B)} is obtained and a representation of S(n, A, B) with the help of a sum of a random number of i.i.d. random variables is derived. How to get asymptotic formulas from local limit theorems is discussed. The special case Stirling numbers of the second kind and its relation to the classical occupancy problem is considered. The Bell numbers and some other related numbers are investigated. Both exact and asymptotic results are derived for the Stirlings and the Bells.