Random digits are collected one at a time until a pattern with given digits is obtained. Blom (1982) and others have determined the mean waiting time for such a pattern. It is proved that when a given pattern has larger mean waiting time than another pattern, then the waiting time for the former is stochastically larger than that for the latter. An application is given to a coin-tossing game.

This paper presents a model for random motion on binary trees. The symmetric model is defined first, and then a model with drift is given. Formulas for absorption probabilities and expected absorption times are presented in both cases.

This paper considers the best-choice problem with a random number of objects having a known distribution. The optimality equation of the problem reduces to an integral equation by a scaling limit. The equation is explicitly solved under conditions on the distribution, which relate to the condition for an OLA policy to be optimal in Markov decision processes. This technique is then applied to three different versions of the problem and an exact value for the asymptotic optimal strategy is found.

We consider the model where φ1, φ2 are real coefficients, not necessarily equal, and the at,'s are a sequence of i.i.d. random variables with mean 0. Necessary and sufficient conditions on the φ 's are given for stationarity of the process. Least squares estimators of the φ 's are derived and, under mild regularity conditions, are shown to be consistent and asymptotically normal. An hypothesis test is given to differentiate between an AR(1) (the case φ1 = φ2) and this threshold model. The asymptotic behavior of the test statistic is derived. Small-sample behavior of the estimators and the hypothesis test are studied via simulated data.

We observe a sequence {Xk}k≧1 of i.i.d. non-negative random variables one at a time. After each observation, we select or reject the observed variable. A variable that is rejected may not be recalled. We want to select N variables as soon as possible subject to the constraint that the sum of the N selected variables does not exceed some prescribed value C > 0. In this paper, we develop a sequential selection procedure that minimizes the expected number of observed variables, and we study some of its properties. We also consider the situation where N → ∞and C/N → α > 0. Some applications are briefly discussed.

Points randomly placed in Rn are independently assigned a speed and direction of motion. The time between collisions with a central n-sphere is obtained. If instead of straight-line motion the points perform Brownian motion, the analogous inter-collision-time distribution is obtained.

Let n points be distributed independently within a k-dimensional unit cube according to density f. At each point, construct a k-dimensional sphere of content an. Let V denote the vacancy, or ‘volume' not covered by the spheres. We derive asymptotic formulae for the mean and variance of V, as n → ∞and an → 0. The formulae separate naturally into three cases, corresponding to nan → 0, nan → a (0 < a <∞) and nan →∞, respectively. We apply the formulae to derive necessary and sufficient conditions for V/E(V) → 1 in L2.

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

The convex hull of n random points chosen independently and uniformly from a d-dimensional ball is a convex polytope. Its expected surface area, its expected mean width and its expected number of facets are explicitly determined.

We investigate sufficient conditions so that is subexponential. Here F is a distribution function on [0, ∞[, with finite mean. Some applications to risk theory and rates of convergence in renewal theory are given.

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

A simple method is proposed for the generation of successive ‘nearest neighbours' to a given origin in an n-dimensional Poisson process. It is shown that the method provides efficient simulation of random Voronoi polytopes. Results are given of simulation studies in two and three dimensions.

In this paper we study the cyclic behaviour of non-homogeneous Markov systems, i.e. the behaviour of the system under the assumption of periodic sequences of transition matrices, input probabilities, output probabilities and total numbers in the system. We provide a general theorem for the limiting structure of such a system under the cyclic behaviour. We also study the asymptotic stability of non-homogeneous Markov systems and theorems are given which characterize asymptotic stability. An application of the above results is given for a British firm, BS.

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 →∞.

We show that the results may be proved in a unified and simple way by using a result of Le Cam.

Consider a file with n records denoted by R1, R2, …, Rn. At each access of a record, the file has to be searched sequentially and it is assumed that the search cost is proportional to the number of probes needed to retrieve the records. The access probabilities p1, p2, …, pn are assumed to be unknown constants and accesses are assumed to be independent. A move-forward self-organizing rule moves a record accessed in the ith position to the lith position without changing the relative ordering of other records where li, = 1, li, < i for i = 2, …, n and li+1≧li. A move-forward rule R is said to be ≦ another move-forward rule R′ if l′i≦li for all i. It is shown that when p2 = p3 = ··· = pn, R ≦ R′, implies cost R ≦ cost R′. This is a generalization of some known results. A new consequence is that the move-up-k + 1 rule is more costly than the move-up-k rule and the move-to-front rule is the most costly of all move-forward rules.

The consistency is developed under mild conditions for the least squares estimator of the parameters of a transfer function time series model.

A method for proving the invariance of hitting-time distributions with respect to the control strategy of a Markov chain is presented. The method is applied to resource-sharing problems. It provides a new proof of a known result and extends it. Sufficient conditions for such an invariance are given and are illustrated by examples.

Conditions under which a function of a finite, discrete-time Markov chain, X(t), is again Markov are given, when X(t) is not irreducible. These conditions are given in terms of an interrelationship between two partitions of the state space of X(t), the partition induced by the minimal essential classes of X(t) and the partition with respect to which lumping is to be considered.

It is shown that, in some particular cases, it is equivalent to characterize a continuous distribution by properties of records and by properties of order statistics. As an application, we give a simple proof that if two successive jth record values and associated to an i.i.d. sequence are such that and are independent, then the sequence has to derive from an exponential distribution (in the continuous case). The equivalence breaks up for discrete distributions, for which we give a proof that the only distributions such that Xk, n and Xk+1, n – Xk, n are independent for some k ≧ 2 (where Xk, n is the kth order statistic of X1, ···, Xn) are degenerate.

Before some random moment θ, independent identically distributed random variables x1, · ··, xθ–1 with common distribution function μ (dx) appear consecutively. After the moment θ, independent random variables xθ, xθ+1, · ·· have another common distribution function f (x)μ (dx). Our information about θ can be constructed only by successively observed values of the x's.

In this paper we find an optimal stopping policy by which we can maximize the probability that the quantity associated with the stopping time is the largest of all θ + m – 1 quantities for a given integer m.