Optical mapping is a new technique to generate restriction maps of DNA easily and quickly. DNA restriction maps can be aligned by comparing corresponding restriction fragment lengths. To relate, organize, and analyse these maps it is necessary to rapidly compare maps. The issue of the statistical significance of approximately matching maps then becomes central, as in BLAST with sequence scoring. In this paper, we study the approximation to the distribution of counts of matched regions of specified length when comparing two DNA restriction maps. Distributional results are given to enable us to compute p-values and hence to determine whether or not the two restriction maps are related. The key tool used is the Chen-Stein method of Poisson approximation. Certain open problems are described.

This note is concerned with integral representations of some transition probabilities of countable state space Markov chains embedded in birth and death processes, of the serial covariances of functions defined on such chains when stationary, and finally the properties of the spectral measure of stationary processes with monotonically decreasing or completely monotonic serial covariances.

Let us consider n stocks with dependent price processes each following a geometric Brownian motion. We want to investigate the American perpetual put on an index of those stocks. We will provide inner and outer boundaries for its early exercise region by using a decomposition technique for optimal stopping.

The coefficients of variation and serial correlation are given for the inter-event intervals of the process obtained by selecting every kth event of the pooled output of several periodic point processes. Relevance to physiological research is explained.

Let {Fn}n ≧ 0 be a sequence of c.d.f. and let {Rn}n ≧ 1 be the sequence of record values in a non-stationary record model where after the (n − 1)th record the population is distributed according to Fn. Then the equidistribution of the nth population and the record increment Rn – Rn– 1 (i.e. Rn – Rn– 1~ Fn) characterizes Fn to have an exponentially decreasing hazard function. To be more precise Fn is the exponential distribution if the support of Rn– 1 generates a dense subgroup in and otherwise the entity of all possible solutions can be obtained in the following way: let for simplicity the above additive subgroup be any c.d.f. F satisfying F(0) = 0, F(1) < 1 can be chosen arbitrarily. Setting λ = – log(1 – F(1)), Fn(x) = 1 – F(x – [x])exp(–λ [x]) is an admissible solution coinciding with F on the interval [0, 1] ([x] denotes the integer part of x). Simple additional assumptions ensuring that Fn is either exponential or geometric are given. Similar results for exponential or geometric tail distributions based on the independence of Rn– 1 and Rn – Rn– 1 are proved.

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

Consider the random walk {Sn} whose summands have the distribution P(X=0) = 1-(2/π), and P(X = ± n) = 2/[π(4n2−1)], for n ≥ 1. This random walk arises when a simple random walk in the integer plane is observed only at those instants at which the two coordinates are equal. We derive the fundamental matrix, or Green function, for the process on the integral [0,N] = {0,1,…,N}, and from this, an explicit formula for the mean time xk for the random walk starting from S0 = k to exit the interval. The explicit formula yields the limiting behavior of xk as N → ∞ with k fixed. For the random walk starting from zero, the probability of exiting the interval on the right is obtained. By letting N → ∞ in the fundamental matrix, the Green function on the interval [0,∞) is found, and a simple and explicit formula for the probability distribution of the point of entry into the interval (−∞,0) for the random walk starting from k = 0 results. The distributions for some related random variables are also discovered.

Applications to stress concentration calculations in discrete lattices are briefly reviewed.

As models for molecular evolution, immune response, and local search algorithms, various authors have used a stochastic process called the evolutionary walk, which goes as follows. Assign a random number to each vertex of the infinite N-ary tree, and start with a particle on the root. A step of the process consists of searching for a child with a higher number and moving the particle there; if no such child exists, the process stops. The average number of steps in this process is asymptotic, as N → ∞, to log N, a result first proved by Macken and Perelson. This paper relates the evolutionary walk to a process called random bisection, familiar from combinatorics and number theory, which can be thought of as a transformed Poisson process. We first give a thorough treatment of the exact walk length, computing its distribution, moments and moment generating function. Next we show that the walk length is asymptotically normally distributed. We also treat it as a mixture of Poisson random variables and compute the asymptotic distribution of the Poisson parameter. Finally, we show that in an evolutionary walk with uniform vertex numbers, the ‘jumps’, ordered by size, have the same asymptotic distribution as the normalized cycle lengths in a random permutation.

Let X be a continuous-time strongly mixing or weakly dependent process and let T be a renewal process independent of X. We show general conditions under which the sampled process $(X_{T_i},T_i-T_{i-1})^{\top}$ is strongly mixing or weakly dependent. Moreover, we explicitly compute the strong mixing or weak dependence coefficients of the renewal sampled process and show that exponential or power decay of the coefficients of X is preserved (at least asymptotically). Our results imply that essentially all central limit theorems available in the literature for strongly mixing or weakly dependent processes can be applied when renewal sampled observations of the process X are at our disposal.

The joint distribution of the time since last exit, and the time until next entrance, into a unique boundary point is given in Formula (1) below. The boundary point may be replaced by a regenerative phenomenon.

Individuals arrive at a geographical area that is initially empty, in accordance with a pure birth process with birth parameters λj = jλ + θ, j ≧ 0. Due to contamination, this geographical area is unsafe for its population and at some fixed time T in the future everyone in the area will be evacuated and no further immigration will be allowed.

Suppose now than an intermediate evacuation time τ, 0 ≦ τ ≦ T, at which time everyone present in the area would be evacuated, is to be chosen. The area would then again fill up with individuals between times τ and T, and, at T, the final evacuation would be made and the area would be permanently sealed off. The problem is to choose τ so as to minimise the total expected cost incurred by time T, where a cost g(x) is incurred for each individual that spends a time x in the area.

We consider the situation when vehicles in a minor road try to cross a major road. The relation between the distributions of the critical, offered and accepted gaps is studied. Estimates of the critical gap distribution from observations of offered and accepted gaps are given and also studied in some simulation examples.

In the study of null-persistent Markov chains the sequence plays an important role. (See for example, Orey (1971), Athreya and Ney (1980).) In this paper we consider the rate of growth of UN to ∞ as N → ∞. This rate of growth is related to the rate of growth of as N → ∞ and also to the rate at which as n → ∞.

We study exponential families within the class of counting processes and show that a mixed Poisson process belongs to an exponential family if and only if it is either a Poisson process or has a gamma structure distribution. This property can be expressed via exponential martingales.

It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, identical in law to the arrival process: this is the analogue of Burke's theorem in this context. In this paper we prove convergence in law to this Brownian motion in a tandem network of Brownian queues: if we have an arbitrary continuous process, satisfying some mild conditions, as an initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

We use an inequality of Sidorenko to show a general relation between local and global subgraph counts and degree moments for locally weakly convergent sequences of sparse random graphs. This yields an optimal criterion to check when the asymptotic behaviour of graph statistics, such as the clustering coefficient and assortativity, is determined by the local weak limit.

As an application we obtain new facts for several common models of sparse random intersection graphs where the local weak limit, as we see here, is a simple random clique tree corresponding to a certain two-type Galton–Watson branching process.

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.

The Bruss–Robertson–Steele (BRS) inequality bounds the expected number of items of random size which can be packed into a given suitcase. Remarkably, no independence assumptions are needed on the random sizes, which points to a simple explanation; the inequality is the integrated form of an $\omega$-by-$\omega$ inequality, as this note proves.