We study the genealogical structure of samples from a population for which any given generation is made up of direct descendants from several previous generations. These occur in nature when there are seed banks or egg banks allowing an individual to leave offspring several generations in the future. We show how this temporal structure in the reproduction mechanism causes a decrease in the coalescence rate. We also investigate the effects of age-dependent neutral mutations. Our main result gives weak convergence of the scaled ancestral process, with the usual diffusion scaling, to a coalescent process which is equivalent to a time-changed version of Kingman's coalescent.

This paper analyzes players’ long-run behavior in an evolutionary model with time-varying mutations under both uniform and local interaction rules. It is shown that a risk-dominant Nash equilibrium in a 2 × 2 coordination game would emerge as the long-run equilibrium if and only if mutation rates do not decrease to zero too fast under both interaction methods. The convergence rates of the dynamic system under both interaction rules are also derived. We find that the dynamic system with local matching may not converge faster than that with uniform matching.

Sankoff and Ferretti (1996) introduced several models of the evolution of chromosome size by reciprocal translocations, where for simplicity they ignored the existence of centromeres. However, when they compared the models to data on six organisms they found that their short chromosomes were too short, and their long chromosomes were too long. Here, we consider a generalization of their proportional model with explicit chromosome centromeres and introduce fitness functions based on recombination probabilities and on the length of the longest chromosome arm. We find a simple formula for the stationary distribution for our model which fits the data on chromosome lengths in many, but not all, species.

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.

The size order problem of the probability state vector elements of a homogeneous Markov system is examined. The time t0 is evaluated, after which the order of the state vector probabilities remains unchanged, and a formula to quickly find a lower bound for t0 is given. A formula for approximating the mode of the state sizes ni(t) as a function of the means Eni(t), and a relation to evaluate P(ni(t) = x+1) by means of certain terms which constitute P(ni(t) = x) are derived.

We consider a controlled M/M/1 queueing system where customers may be subject to two potential rejections. The first occurs upon arrival and is dependent on the number of customers in the queue and the service rate of the customer currently in service. The second, which may or may not occur, occurs immediately prior to the customer receiving service. That is, after each service completion the customer in the front of the queue is assessed and the service rate of that customer is revealed. If the second decision-maker recommends rejection, the customer is denied service with a fixed probability. We show the existence of long-run average optimal monotone switching-curve policies. Further, we show that the average reward is increasing in the probability that the second decision-maker's recommendation of rejection is honored. Applications include call centers with delayed classifications and manufacturing systems when the server is responsible for multiple tasks.

We consider a multicomponent maintenance system controlled by an age replacement policy: when one of the components fails, it is immediately replaced; all components older than a threshold age θ are preventively replaced. Costs are associated with each maintenance action, such as replacement after failure or preventive replacement. We derive a weak derivative estimator for the derivative of the cost performance with respect to θ. The technique is quite general and can be applied to many other threshold optimization problems in maintenance. The estimator is easy to implement and considerably increases the efficiency of a Robbins-Monro type of stochastic approximation algorithm. The paper is self-contained in the sense that it includes a proof of the correctness of the weak derivative estimation algorithm.

Univariate probability inequalities have received extensive attention. It has been shown that under certain conditions, product-type bounds are valid and sharper than summation-type bounds. Although results concerning multivariate inequalities have appeared in the literature, product-type bounds in a multivariate setting have not yet been studied. This note explores an approach using graph theory and linear programming techniques to construct product-type lower bounds for the probability of the intersection among unions of k sets of events.

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

In this paper we study random variables related to a shock reliability model. Our models can be used to study systems that fail when k consecutive shocks with critical magnitude (e.g. above or below a certain critical level) occur. We obtain properties of the distribution function of the random variables involved and we obtain their limit behaviour when k tends to infinity or when the probability of entering a critical set tends to zero. This model generalises the Poisson shock model.

Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

We investigate an optimal stopping time problem which arises from pricing Russian options (i.e. perpetual look-back options) on a stock whose price fluctuations are modelled by adjoining a hidden Markov process to the classical Black-Scholes geometric Brownian motion model. By extending the technique of smooth fit to allow jump discontinuities, we obtain an explicit closed-form solution. It gives a non-standard application of the well-known smooth fit principle where the optimal strategy involves jumping over the optimal boundary and by an arbitrary overshoot. Based on the optimal stopping analysis, an arbitrage-free price for Russian options under the hidden Markov model is derived.

Let X = (X(t):t ≥ 0) be a Lévy process and X∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X∊(1)). In simulation, X - X∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

We consider Markov chains in the context of iterated random functions and show the existence and uniqueness of an invariant distribution under a local contraction condition combined with a drift condition, extending results of Diaconis and Freedman. From these we deduce various other topological stability properties of the chains. Our conditions are typically satisfied by, for example, queueing and storage models where the global Lipschitz condition used by Diaconis and Freedman normally fails.

Let Xn, n ≥ 1 be a sequence of trials taking values in a given set A, let ∊ be a pattern (simple or compound), and let Xr,∊ be a random variable denoting the waiting time for the rth occurrence of ∊. In the present article a finite Markov chain imbedding method is developed for the study of Xr,∊ in the case of the non-overlapping and overlapping way of counting runs and patterns. Several extensions and generalizations are also discussed.

We study the convergence of certain matrix sequences that arise in quasi-birth-and-death (QBD) Markov chains and we identify their limits. In particular, we focus on a sequence of matrices whose elements are absorption probabilities into some boundary states of the QBD. We prove that, under certain technical conditions, that sequence converges. Its limit is either the minimal nonnegative solution G of the standard nonlinear matrix equation, or it is a stochastic solution that can be explicitly expressed in terms of G. Similar results are obtained relative to the standard matrix R that arises in the matrix-geometric solution of the QBD. We present numerical examples that clarify some of the technical issues of interest.

In this paper two burn-in procedures for a general failure model are considered. There are two types of failure in the general failure model. One is Type I failure (minor failure) which can be removed by a minimal repair or a complete repair and the other is Type II failure (catastrophic failure) which can be removed only by a complete repair. During a burn-in process, with burn-in Procedure I, the failed component is repaired completely regardless of the type of failure, whereas, with burn-in Procedure II, only minimal repair is done for the Type I failure and a complete repair is performed for the Type II failure. In field use, the component is replaced by a new burned-in component at the ‘field use age’ T or at the time of the first Type II failure, whichever occurs first. Under the model, the problems of determining optimal burn-in time and optimal replacement policy are considered. The two burn-in procedures are compared in cases when both the procedures are applicable.

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

Daley and Vesilo (1997) introduced long-range count dependence (LRcD) for stationary point processes on the real line as a natural augmentation of the classical long-range dependence of the corresponding interpoint sequence. They studied LRcD for some renewal processes and some output processes of queueing systems, continuing the previous research on such processes of Daley (1968), (1975). Subsequently, Daley (1999) showed that a necessary and sufficient condition for a stationary renewal process to be LRcD is that under its Palm measure the generic lifetime distribution has infinite second moment. We show that point processes dominating, in a sense of stochastic ordering, LRcD point processes are LRcD, and as a corollary we obtain that for arbitrary stationary point processes with finite intensity a sufficient condition for LRcD is that under Palm measure the interpoint distances are positively dependent (associated) with infinite second moment. We give many examples of LRcD point processes, among them exchangeable, cluster, moving average, Wold, semi-Markov processes and some examples of LRcD point processes with finite second Palm moment of interpoint distances. These examples show that, in general, the condition of infiniteness of the second moment is not necessary for LRcD. It is an open question whether the infinite second Palm moment of interpoint distances suffices to make a stationary point process LRcD.

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.