This paper is concerned with Markov chain models for flows of a finite population among a set of groups, where the individuals base their decisions on which group to go next partially on the current frequency distribution (profile). For a certain class of these models, the transition matrix of the profile process is analyzed algebraically, leading to surprisingly simple asymptotic results. Furthermore, in a model with after effects, the absorption probabilities are derived.

We derive a law of large numbers for a tagged particle in the one-dimensional totally asymmetric simple exclusion process under a scaling different from the usual Euler scaling. By interpreting the particles as the servers of a series of queues we use this result to verify an open conjecture about the scaling behavior of the departure times from a long series of queues.

We study the behavior of large loss networks in which the offered traffic is subject to acceptance controls. Hunt and Kurtz proved a functional law of large numbers for the dynamics of such networks, as capacity and offered traffic are allowed to increase in proportion. However, limiting dynamics were not in general uniquely identified. We establish further results identifying these dynamics under given conditions. We also investigate the existence of fixed points for these dynamics and relate them to limiting equilibrium behavior, permitting the investigation of common modelling assumptions. We study in detail single- and two-resource networks and we give an example of bistability for the former.

In 1977 Vershik and Kerov deduced the asymptotic normalized length of the longest increasing sequence among independent points uniformly distributed on the unit square. We solve the analogous problem for points on the planar square lattice that are present independently of each other.

This paper investigates an alternative way of using the Stein-Chen method in Poisson approximations. There are three principal bounds stated in terms of reduced Palm probabilities for general point processes. The first two are for the accuracy of Poisson random variable approximation to the distribution of the number of points in a point process with respect to the total variation metric and the Wasserstein metric, and the third is for bounding the errors of Poisson process approximation to the distribution of a point process on a general compact space with respect to a Wasserstein metric. The bounds are frequently sharper than the previous results using the Stein-Chen method when the expected number of points is large.

We give an approximation for optimal filtering estimates of a diffusion type signal and its observation under contamination affecting their drifts. The method of analysis is based on the averaging principle and convergence in total variation for distributions of signal and observation.

Let W be a random variable with mean zero and variance $\sigma^2$. The distribution of a variate $W^*$, satisfying $EWf(W) = \sigma^2 Ef'(W^*)$ for smooth functions f , exists uniquely and defines the zero bias transformation on the distribution of W. The zero bias transformation shares many interesting properties with the well-known size bias transformation for nonnegative variables, but is applied to variables taking on both positive and negative values. The transformation can also be defined on more general random objects. The relation between the transformation and the expression $wf'(w) - \sigma^2 f''(w)$ which appears in the Stein equation characterizing the mean zero, variance $\sigma^2$ normal $\sigma Z$can be used to obtain bounds on the difference $E{h(W/ \sigma) - h(Z)}$ for smooth functions h by constructing the pair $(W, W^*)$ jointly on the same space. When W is a sum of n not necessarily independent variates, under certain conditions which include a vanishing third moment, bounds on this difference of the order $1/n$ for classes of smooth functions h may be obtained. The technique is illustrated by an application to simple random sampling.

It is known that maxima of independent Poisson variables cannot be normalized to converge to a nondegenerate limit distribution. On the other hand, the Normal distribution approximates the Poisson distribution for large values of the Poisson mean, and maxima of random samples of Normal variables may be linearly scaled to converge to a classical extreme value distribution. We here explore the boundary between these two kinds of behavior. Motivation comes from the wish to construct models for the statistical analysis of extremes of background gamma radiation over the United Kingdom. The methods extend to row-wise maxima of certain triangular arrays, for which limiting distributions are also derived.

Consider a rooted labelled tree graph $\tau_n$ having a total of n vertices. The width function counts the number of vertices as a function of the distance to the root $\phi$. In this paper we compute large n asymptotic behavior of the width functions for two classes of tree graphs (both random and deterministic) of the following types: (i) Galton-Watson random trees $\tau_n$ conditioned on total progeny and (ii) a class of deterministic self-similar trees which include an "expected" Galton-Watson tree in a sense to be made precise. The main results include: (i) an extension of Aldous's theorem on "search-depth" approximations by Brownian excursion to the case of weighted Galton-Watson trees; (ii) a probabilistic derivation which generalizes previous results by Troutman and Karlinger on the asymptotic behavior of the expected width function and provides the fluctuation law; and (iii) width function asymptotics for a class of deterministic self-similar trees of interest in the study of river network data.

Let ${X_i: i \geq 1}$ be i.i.d. with uniform distribution $[- 1/2, 1/2]^d, d \geq 2$, and let $T_n$ be a minimal spanning tree on ${X_1, \dots, X_n}$. For each strictly positive integer $\alpha$, let $N({X_1, \dots, X_n}; \alpha)$ be the number of vertices of degree $\alpha$ in $T_n$. Then, for each $\alpha$ such that $P(N({X_1, \dots, X_{\alpha+1}}; \alpha) = 1) > 0$, we prove a central limit theorem for $N({X_1, \dots, X_n}; \alpha)$.

We study the time it takes until a fluid queue with a finite, but large, holding capacity reaches the overflow point. The queue is fed by an on/off process with a heavy tailed on distribution which is known to have long memory. It turns out that the expected time until overflow, as a function of capacity L, increases only polynomially fast; so overflows happen much more often than in the "classical" light tailed case, where the expected over-flow time increases as an exponential function of L. Moreover, we show that in the heavy tailed case overflows are basically caused by single huge jobs. An implication is that the usual $GI/G/1$ queue with finite but large holding capacity and heavy tailed service times will overflow about equally often no matter how much we increase the service rate. We also study the time until overflow for queues fed by a superposition of k iid on/off processes with a heavy tailed on distribution, and we show the benefit of pooling the system resources as far as time until overflow is concerned.

Trunk reservation is a simple, robust and extremely effective mechanism for controlling loss systems which allows priority to be given to chosen traffic streams. We consider the control of a single resource under a limiting regime in which capacity and arrival rates increase together. We obtain trunk reservation control policies which are asymptotically optimal when calls have differing capacity requirements, holding times, arrival rates and reward rates. The priority levels associated with these trunk reservation policies arise from an attainable bound on the performance of any control policy.

This paper deals with rates of convergence in the CLT for certain types of dependency. The main idea is to combine a modification of a theorem of Stein, requiring a coupling construction, with a dynamic set-up provided by a Markov structure that suggests natural coupling variables. More specifically, given a stationary Markov chain $X^{(t(}$, and a function $U = U(X^{(t)})$, we propose a way to study the proximity of U to a normal random variable when the state space is large.

We apply the general method to the study of two problems. In the first, we consider the antivoter chain $X^{(t)} = {X_i^{(t)}} _{i \epsilon \mathscr{V}}, t = 0, 1, \dots,$ where $\mathscr{V}$ is the vertex set of an n-vertex regular graph, and $X_i^{(t)} = +1 \text{or} -1$. The chain evolves from time t to $t + 1$ by choosing a random vertex i, and a random neighbor of it j, and setting $X_i^{(t+1)} = -X_j^{(t)}$ and $X_k^{(t+1)} = X_k^{(t)}$ for all $k \not= i$. For a stationary antivoter chain, we study the normal approximation of $U_n = U_n^{(t)} = \Sigma_i X_i^{(t)}$ for large n and consider some conditions on sequences of graphs such that $U_n$ is asymptotically normal, a problem posed by Aldous and Fill.

The same approach may also be applied in situations where a Markov chain does not appear in the original statement of a problem but is constructed as an auxiliary device. This is illustrated by considering weighted U-statistics. In particular we are able to unify and generalize some results on normal convergence for degenerate weighted U-statistics and provide rates.

Let $X_i$ and $Y_i$ follow noncentral chi-square distributions with the same degrees of freedom $\nu_i$ and noncentrality parameters $\delta_i^2$ and $\mu_i^2$, respectively, for $i = 1, \dots, n$, and let the $X_i$'s be independent and the $Y_i$'s independent. A necessary and sufficient condition is obtained under which $\Sigma_{i = 1}^n \lambda_i X_i$ is stochastically smaller than $\Sigma_{i = 1}^n \lambda_i Y_i$ for all nonnegative real numbers $\lambda_i \geq \dots \geq \lambda_n$. Reformulating this as a result in geometric probability, solutions are obtained, in particular, to the problems of monotonicity and location of extrema of the probability content of a rotated ellipse under the standard bivariate Gaussian distribution. This complements results obtained by Hall, Kanter and Perlman who considered the behavior of the probability content of a square under rotation. More generally, it is shown that the vector of partial sums $(X_1, X_1 + X_2, \dots, X_1 + \dots + X_n)$ is stochastically smaller than $(Y_1, Y_1 + Y_2, \dots, Y_1 + \dots + Y_n)$ if and only if $\Sigma_{i=1}^n \lambda_i X_i$ is stochastically smaller than $\Sigma_{i=1}^n \lambda_i Y_i$ for all nonnegative real numbers $\lambda_1 \geq \dots \geq \lambda_n$.

We consider the annealing diffusion process and investigate convergence rates. Namely, the diffusion $dX_t = -\nabla V(X_t)dx+\sigma(t)dB_t$, where $(B_t)_t\leq 0$ is the $d$-dimensional Brownian motion and $\sigma(t)$ decreases to zero, we prove a large deviation principle for $(V(X_t))$ and weak convergence of $(\sigma^{-2}(t)(V(X_t)-\inf V)).$