The Ewens sampling formula is a family of measures on permutations, that arises in population genetics, Bayesian statistics and many other applications. This family is indexed by a parameter $\theta > 0$; the usual uniform measure is included as the special case $\theta = 1$. Under the Ewens sampling formula with parameter $\theta$, the process of cycle counts $(C_1(n), C_2(n), \ldots, C_n(n), 0, 0, \ldots)$ converges to a Poisson process $(Z_1, Z_2, \ldots)$ with independent coordinates and $\mathbb{E}Z_j = \theta/j$. Exploiting a particular coupling, we give simple explicit upper bounds for the Wasserstein and total variation distances between the laws of $(C_1(n), \ldots, C_b(n))$ and $(Z_1, \ldots, Z_b)$. This Poisson approximation can be used to give simple proofs of limit theorems with bounds for a wide variety of functionals of such random permutations.

In this paper Stein's method is considered in the context of approximation by a multinomial distribution. By using a probabilistic argument of Barbour, whereby the essential ingredients necessary for the application of Stein's method are derived, the Stein equation for the multinomial distribution is obtained. Bounds on the smoothness of its solution are derived and are used in three examples to give error bounds for the multinomial approximation to the distribution of a random vector.

A light traffic limit theorem is proved for random walks in a triangular array setting similar to the heavy traffic situation, the basic assumption being on the moments in the right tail of the increment distribution. When specialized to GI/G/1 queues, this result is shown to contain the known types of light traffic behaviour in this setting (Daley and Rolski) as well as some additional ones. Intuitively, the results state that typically delay in light traffic occurs with just one customer in the system, and then as a result of long service times and/or short interarrival times in a balance which depends on the particular parameters of the model. Particular attention is given to queues with phase-type service times, for example of Coxian type.

We consider a multitype branching process in which particles move according to probability laws that depend on the type of a particle. The composition of the population, namely the proportions of numbers of particles of different types, varies over the region of motion. In this paper, we study the growth rate and population composition of the process throughout its domain of supercriticality.

The skip list was introduced by Pugh in 1989 as a data structure for dictionary operations. Using a binary tree representation of skip lists, we obtain the limit law for the path lengths of the leaves in the skip list. We also show that the height (maximal path length) of a skip list holding $n$ elements is in probability asymptotic to $c \log_{1/p} n$, where $c$ is the unique solution greater than 1 of the equation $\log(1 - p) = \log(c - 1) - \lbrack c/(c - 1) \rbrack \log c$, and $p \in (0, 1)$ is a design parameter of the skip list.

In this paper we prove an optimal spectral structure theorem for stochastic matrices reversible with respect to a fixed probability measure $\pi$ on a finite set and devise a new simulation algorithm for Markov random fields. We compute the minimum value for the second largest eigenvalue of all such matrices and characterize the class of matrices for which this minimum is attained. In fact, they share a common right eigenvector that can be written in terms of $\pi$. Furthermore, by iterating this procedure, we obtain a unique matrix which is minimal with respect to the lexicographic order of the eigenvalues. We give a probabilistic interpretation of the corresponding eigenvectors. Our results allow us to devise a dynamic Monte Carlo scheme which has an optimal worst-case performance. Regarding the simulation of lattice-based Gibbs distributions, we design a modified Gibbs sampler, whose performance is better in terms of both weak convergence at low temperatures and asymptotic variance of time averages at all temperatures.

We compute the resolvent of the degenerate, two-dimensional diffusion process introduced by Benes, Karatzas and Rishel in the study of a stochastic control problem with partial observations. The explicit nature of our computations allows us to show that this diffusion can be constructed uniquely (in the sense of the probability law) starting at any point on the plane, including the origin, and to solve explicitly the control problem of Benes, Karatzas and Rishel for very general cost functions. Our derivation combines probabilistic techniques, with use of the so-called "principle of smooth fit."

A law of large numbers and a central limit theorem are proved for a locally interacting particle system. This system describes a chemical reaction with diffusion with linear creation and quadratic annihilation of particles. The deterministic limit is the solution of a nonlinear reaction-diffusion equation defined on an $n$-dimensional unit cube. The law of large numbers holds for any dimension $n$ and arbitrary times, whereas the central limit theorem holds only for dimension $n \leq 3$ and on a certain bounded time interval (depending on the initial distribution and on the creation rate). A propagation of chaos expansion of the correlation functions is used.

Consider a measure $\mu$ on $\lbrack 0, 1\rbrack^2$, and $2n$ points $X_1, \cdots, X_n, Y_1, \cdots, Y_n$ that are independent and distributed according to $\mu$. Consider $2n$ points $U_1, \cdots, U_n, V_1, \cdots, V_n$ that are independent and uniformly distributed on $\lbrack 0, 1 \rbrack$. Then there exists a constant $K$ (independent of $\mu$) such that if $s \leq \sqrt n / K$, with probability close to 1 we can find a one-to-one map $\pi$ from $\{1, \cdots, n\}$ to itself such that $\forall i \leq n, \quad |U_i - V_{\pi(i)}| \leq \frac{K}{s},$ $\frac{1}{n} \sum_{i \leq n} |X_i - Y_{\pi(i)}| \leq K \big(\frac{s}{n}\big)^{1/2}.$

This paper presents some results that are useful in the study of asymptotic approximations of boundary crossing probabilities for random walks. The main result is a refinement of an asymptotic expansion of Siegmund concerning moments of the first ladder height of random walks having small positive drift. An analysis of the covariance between the first passage time and the overshoot of a random walk over a horizontal boundary contributes to the development of the main result and is of independent interest as well. An application of these results to a "moderate deviations" approximation for the probability distribution of the time to false alarm in the cusum procedure is briefly described.

This paper studies coupling methods for proving convergence in distribution of (typically Markovian) stochastic processes in continuous time to their stationary distribution. The paper contains: (a) a simple lemma on $\varepsilon$-coupling; (b) conditions for Markov processes to couple in compact sets; (c) new variants of the coupling proof of the renewal theorem; (d) a convergence result for stochastically monotone Markov processes in an ordered Polish space; and (e) a case study of a queue with superposed renewal input. In a companion paper with Foss, similar discussion is given for many-server queues in continuous time.

We derive some results which may be helpful to buyers of software testing for faults, or to buyers of large lots screening for defectives. Suppose that a fixed but unknown number $n$ of faults or defectives remain before testing. In the testing phase they are observed at random times, $X_1, X_2, \cdots, X_n$, which are order statistics corresponding to $n$ i.i.d. random variables. Since testing is usually an ongoing activity, this distribution is typically known. Under this assumption we derive a stopping criterion that guarantees, for any specified level $\alpha$ and integer $m$, that for all $n > m$, with probability exactly $1 - \alpha$, when stopping occurs, the software has no more than $m$ faults remaining. We study various properties of this stopping rule, both finite and asymptotic, and show that it is optimal in a certain sense. We modify a conservative stopping rule proposed by Marcus and Blumenthal to make it exact, and we give some numerical comparisons.