This is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite state spaces and we are able to give a self-contained development. Examples of applications include the study of a Metropolis chain for the binomial distribution, sharp results for natural chains on the box of side n in d dimensions and improved rates for exclusion processes. We also show that for most r-regular graphs the log-Sobolev constant is of smaller order than the spectral gap. The log-Sobolev constant of the asymmetric two-point space is computed exactly as well as the log-Sobolev constant of the complete graph on n points.

This paper studies the instability of multiclass queueing networks. We prove that if a fluid limit model of the queueing network is weakly unstable, then the queueing network is unstable in the sense that the total number of customers in the queueing network diverges to infinity with probability 1 as time $t \to \infty$. Our result provides a converse to a recent result of Dai which states that a queueing network is positive Harris recurrent if a corresponding fluid limit model is stable. Examples are provided to illustrate the usage of the result.

Dupuis and Williams proved that a sufficient condition for the positive recurrence and the existence of a unique stationary distribution for a semimartingale reflecting Brownian motion in an orthant (SRBM) is that all solutions of an associated deterministic Skorohod problem are attracted to the origin. In this paper, we derive a sufficient condition under which we can construct an explicit linear Lyapunov function for the Skorohod problem. Thus, this implies a sufficient condition for the stability of the deterministic Skorohod problem. The existence of such a linear Lyapunov function is equivalent to the feasibility of a set of linear inequalities. In the two-dimensional case, we recover the necessary and sufficient conditions for the positive recurrence. Some explicit sufficient conditions are derived for the higher-dimensional case.

In this paper we consider the asymptotic time average variance of the actual waiting time of the PH/PH/1 queue. To this end Poisson's equation plays a major role, since the solution to Poisson's equation is shown to be intimately connected with the desired variance. We derive explicit solutions to Poisson's equation using phase-type methodology, and present some numerical examples.

Here is a new genetic algorithm. It is built by randomly perturbing a two operator crossover-selection scheme. Three conditions of biological relevance are imposed on the crossover. A new selection mechanism is used, which has the decisive advantage of preserving the diversity of the individuals in the population. The attractors of the unperturbed process are particular equifitness subsets of populations endowed with a rich structure. The random vanishing perturbations are twofold: local perturbations of the individuals (mutations) and loosening of the selection pressure. When the population size is greater than a critical value which depends strongly on the optimization problem, their delicate asymptotic interaction ensures the convergence (possibly in finite time) of the population to the ideal attractor whose populations contain all the maxima of the fitness function. The process explores without respite the neighborhoods of the best points found so far (instead of focusing on a particular point) and finds simultaneously all the global maxima of the fitness function; it seems to be the first cooperative search procedure of this kind.

In this paper we construct a stochastic particle method for the Burgers equation with a monotone initial condition; we prove that the convergence rate is $O(1/ \sqrt{N} + \sqrt{\Delta t})$ for the $L^1 (\mathbb{R} \times \Omega)$ norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the difficulty of the analysis comes from the discontinuity of the interaction kernel, which is equal to the Heaviside function.

In a previous paper we showed how the algorithm and the result extend to the case of nonmonotone initial conditions for the Burgers equation. We also treated the case of nonlinear PDE's related to particle systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the two-dimensional inviscid Navier-Stokes equation.

We consider optimal stopping problems with loss function q depending on the rank of the stopped random variable. Samuels asked whether there exists an exchangeable sequence of random variables $X_1, \dots, X_n$ without ties for which the observation of the values of the $X_i$'s gives no advantage in comparison with the observation of just the relative ranks of the variables. We call distributions of the sequences with this property q-noninformative and derive necessary and sufficient conditions for this property. Extending an impossibility result of B. Hill, we show that, for any $n > 1$, there are certain losses q for which q-noninformative distributions do not exist. Special attention is given to the classical problem of minimizing the expected rank: for n even we construct explicitly universal randomized stopping rules which are strictly better than the rank rules for any exchangeable sequence.

We consider a stochastic model for the spread of an epidemic in a population split into m groups in which both infective and susceptible individuals are able to move between groups. Using a coupling argument similar to those applied to various other epidemic models by previous authors, we show that as the initial susceptible population becomes large, the process of infectives in this epidemic model converges to a multitype birth-and-death process with time-dependent birth rates. The behavior of this limiting process is then considered, in particular, the conditions under which extinction is almost certain.

Let ${Z_n}$ be a branching process whose offspring distributions vary with n. It is shown that the sequence ${\max_{i>0} P(Z_n = i)}$ has a limit. Denote this limit by M. It turns out that M is positive only if the offspring variables rapidly approach constants. Let ${c_n}$ be a sequence of constants and $W_n = Z_n / c_n$. It will be proven that $M = 0$ is necessary and sufficient for the limit distribution functions of all convergent ${W_n}$ to be continuous on $(0, \infty)$. If $M > 0$ there is, up to an equivalence, only one sequence ${c_n}$ such that ${W_n}$ has a limit distribution with jump points in $(0, \infty)$. Necessary and sufficient conditions for continuity of limit distributions are derived in terms of the offspring distributions of ${Z_n}$.

The model considered here has arisen in a number of completely separate contexts: release of neurotransmitter at neuromuscular synapses, unravelling of strands of DNA, differentiation of cells into heterocysts in algae and growth of crystals. After a shear transformation the model becomes a Markov process, based on a Poisson process on the upper half plane, homogeneous in the horizontal (time) direction, which increases at unit rate except for occasional "drops." By considering the process separately when it is above or below a given "level," for any interval on the time axis, we obtain in particular exact moment results and prove asymptotic normality for long time intervals for the number of downcrossings in the interval, the total time in the interval when the process is below the specified level and the number of drops in the interval. Limit distributions as the length of interval tends to infinity are obtained for the level at which the interval is "covered." It is shown that several problems considered in the literature have analytic solutions as special cases of the general model. The numerical results from one special case are compared to statistics obtained from experimental data from neurobiology.

We consider the bisexual Galton-Watson process (BGWP) with promiscuous mating, that is, a branching process which behaves like an ordinary Galton-Watson process as long as at least one male is produced in each generation. For the case of Poissonian reproduction, it was pointed out by Daley, Hull and Taylor that the extinction probability of such a BGWP apparently behaves like a constant times the respective probability of its asexual counterpart (where males do not matter) providing the number of ancestors grows to infinity. They further mentioned that they had no theoretical justification for this phenomenon. In the present article we will prove upper and lower bounds for the ratio between the two extinction probabilities and introduce a recursive algorithm that can easily be implemented on a computer to produce very accurate approximations for that ratio. The final section contains a number of numerical results that have been obtained by use of this algorithm.

In this paper we study numerical methods to approximate the adapted solutions to a class of forward-backward stochastic differential equations (FBSDE's). The almost sure uniform convergence as well as the weak convergence of the scheme are proved, and the rate of convergence is proved to be as good as the approximation for the corresponding forward SDE. The idea of the approximation is based on the four step scheme for solving such an FBSDE, developed by Ma, Protter and Yong. For the PDE part, the combined characteristics and finite difference method is used, while for the forward SDE part, we use the first order Euler scheme.

Functionals of a two-parameter integrated periodogram have been used for detecting a change in the spectral distribution of a stationary sequence. The bases for these results are functional central limit theorems for the integrated periodogram with a Gaussian limit field. We prove functional central limit theorems for a general linear sequence having a finite fourth moment which is shown to be the optimal moment condition. Our approach is via an approximation of the integrated periodogram by a finite linear combination of sample autocovariances. This gives special insight into the structure of the Gaussian limit field.

Let $(X_i, Y_i), i = 1, \dots, n$, be independent random vectors with a standard bivariate normal distribution and let $s_X$ and $s_Y$ be the sample standard deviations. For arbitrary $p, 0 < p < 1$, define $T_i = pX_i + (1 - p) Y_i$ and $Z_i = pX_i / s_X + (1 - p) Y_i / s_Y, i = 1, \dots, n$. The couple of pairs $(T_i, Z_i)$ and $(T_j, Z_j)$ is said to be discordant if either $T_i < T_j$ and $Z_i > Z_j$ or $T_i > T_j$ and $Z_i < Z_j$. It is shown that the expected number of discordant couples of pairs is asymptotically equal to $n^{3/2}$ times an explicit constant depending on p and the correlation coefficient of $X_i$ and $Y_i$. By an application of the Durbin-Stuart inequality, this implies an asymptotic lower bound on the expected value of the sum of $(\rank(Z_i) - \rank(T_i))^+$. The problem arose in a court challenge to a standard procedure for the scoring of multipart written civil service examinations. Here the sum of the positive rank differences represents a measure of the unfairness of the method of scoring.

We show that the stochastic versions of some classical problems in combinatorial optimization may be imbedded in multiparameter subadditive processes having an intrinsic ergodic structure. A multiparameter generalization of Kingman's subadditive ergodic theorem is used to capture strong laws for these optimization problems, including the traveling salesman and minimal spanning tree processes. In this way we make progress on some open problems and provide alternate proofs of some well known asymptotic results.

We express Gittins indices for multi-armed bandit problems as Laurent expansions around discount factor 1. The coefficients of these expan-sions are then used to characterize stationary optimal policies when the optimality criteria are sensitive-discount optimality (otherwise known as Blackwell optimality), average-reward optimality and average-overtaking optimality. We also obtain bounds and derive optimality conditions for policies of a type that continue playing the same bandit as long as the state of that bandit remains in prescribed sets.

This paper deals with the distribution function of the sojourn time of Brownian motion with drift. By some recent results of J. Akahori and A. Dassios this distribution function can be expressed in the form of a double integral. In this paper it is shown that the distribution function of the sojourn time can be expressed simply as a single integral. The result obtained is a generalization of the arc-sine law of Paul Lévy.

The purpose of this note is to obtain a representation of the distribution of the $\alpha$-quantile of a process with stationary and independent increments as the sum of the supremum and the infimum of two rescaled independent copies of the process. This representation has already been proved for a Brownian motion. The proof is based on already known discrete time results.