Large deviation principles are established for some random combinatorial structures including the Ewens sampling formula and the Pitman sampling formula. A path-level large deviation principle is established for the former on the cadlag space D$(o, 1], R)$ equipped with the uniform convergence topology, and the rate function is the same as for a Poisson process justifying the Poisson process approximation for the Ewens sampling formula at the large deviation level. A large deviation principle for the total number of parts in a partition is obtained for the Pitman formula; here the rate function depends only on one of the two parameters which display the different roles of the two parameters at different scales. In addition to these large deviation results, we also provide an embedding scheme which gives the Pitman sampling formula. A product of this embedding is an intuitive alternate proof of a result of Pitman on the limiting total number of parts.

We show, under regularity conditions, that a nonnegative nondecreasing real-valued stochastic process satisfies a large deviation principle (LDP) with nonlinear scaling if and only if its inverse process does. We also determine how the associated scaling and rate functions must be related. A key condition for the LDP equivalence is for the composition of two of the scaling functions to be regularly varying with nonnegative index. We apply the LDP equivalence to develop equivalent characterizations of the asymptotic decay rate in nonexponential asymptotics for queue-length tail probabilities. These alternative characterizations can be useful to estimate the asymptotic decay constant from systems measurements.

We consider a single class, acyclic network of $G/G/1$ queues. We impose some mild assumptions on the service and external arrival processes and we characterize the large deviations behavior of all the processes resulting from various operations in the network. For the network model that we are considering, these operations are passing-through-a-single-server-queue (the process resulting from this operation being the departure process), superposition of independent processes and deterministic splitting of a process into a number of processes. We also characterize the large deviations behavior of the waiting time and the queue length observed by a typical customer in a single server queue. We prove that the assumptions imposed on the external arrival processes are preserved by these operations, and we show how to apply inductively these results to obtain the large deviations behavior of the waiting time and the queue length in all the queues of the network. Our results indicate how these large deviations occur, by concretely characterizing the most likely path that leads to them.

Using large deviation techniques, we analyze the tail behavior of the stationary distribution of the buffer content process for a two-station communication network. We also show how the associated rate function can be expressed as the solution to a finite-dimensional variational problem. Along the way, we develop a number of results and techniques that are of independent interest, including continuity results for the input-output mapping for certain multiclass fluid models and a new technique for obtaining large deviation principles for invariant distributions from sample path large deviation results.

We study uniform acceleration (UA) expansions of finite-state continuous-time Markov chains with time-varying transition rates. The UA expansions can be used to justify, evaluate and refine the pointwise stationary approximation, which is the steady-state distribution associated with the time-dependent generator at the time of interest. We obtain UA approximations from these UA asymptotic expansions. We derive a time-varying analog to the uniformization representation of transition probabilities for chains with constant transition rates, and apply it to establish asymptotic results related to the UA asymptotic expansion. These asymptotic results can serve as appropriate time-varying analogs to the notions of stationary distributions and limiting distributions. We illustrate the UA approximations by doing a numerical example for the time-varying Erlang loss model.

A popular estimator of the index of regular variation in heavy-tailed models is Hill's estimator. We discuss the consistency of Hill's estimator when it is applied to certain classes of heavy-tailed stationary processes. One class of processes discussed consists of processes which can be appropriately approximated by sequences of m-dependent random variables and special cases of our results show the consistency of Hill's estimator for (i) infinite moving averages with heavy-tail innovations, (ii) a simple stationary bilinear model driven by heavy-tail noise variables and (iii) solutions of stochastic difference equations of the form $$Y_t = A_tY_{t-1} + Z_t, -\infty < t < \infty$$ where ${(A_n, Z_n), \quad -\infty < n < \infty}$ are iid and the Z's have a regularly varying tail probabilities. Another class of problems where our methods work successfully are solutions of stochastic difference equations such as the ARCH process where the process cannot be successfully approximated by m-dependent random variables. A final class of models where Hill estimator consistency is proven by our tail empirical process methods is the class of hidden semi-Markov models.

Consider a function $f: B \to R_'$, where B is a compact subset of $R^m$ and consider a "simulation" used to estimate $f(x), x \epsilon B$ with the following properties: the simulation can switch from one $x \epsilon B$ to another in zero time, and a simulation at x lasting t units of time yields a random variable with mean $f(x)$ and variance $v(x)/t$ . With such a simulation we can divide T units of time into as many separate simulations as we like. Therefore, in principle we can design an "experiment" that spends $\tau(A)$ units of time simulating points in each $A \epsilon \mathscr{B}$, where $\mathscr{B}$ is the Borel $\sigma$-field on B and $\tau$ is an arbitrary finite measure on $(B, \mathscr{B})$. We call a design specified by a measure $\tau$ a "generalized design." We propose an approximation for f based on the data from a generalized design. When $\tau$ is discrete, the approximation, $\hat{f}$, reduces to a "Kriging"-like estimator. We study discrete designs in detail, including asymptotics (as the length of the simulation increases) and a numerical procedure for finding optimal n-point designs based on a Bayesian interpretation of $\hat{f}$ . Our main results, however, concern properties of generalized designs. In particular, we give conditions for integrals of $\hat{f}$ to be consistent estimates of the corresponding integrals of f. These conditions are satisfied for a large class of functions, f , even when $v(x)$ is not known exactly. If f is continuous and $\tau$ has a density, then consistent estimation of $f(x), x \epsilon B$ is also possible. Finally, we use the Bayesian interpretation of $\hat{f}$ to derive a variational problem satisfied by globally optimal designs. The variational problem always has a solution and we describe a sequence of n-point designs that approach (with respect to weak convergence) the set of globally optimal designs. Optimal designs are calculated for some generic examples. Our numerical studies strongly suggest that optimal designs have smooth densities.

We analyze the storage capacity of the Hopfield model with correlated patterns $(\xi_i^{\nu})$. We treat both the case of semantically and spatially correlated patterns (i.e., the patterns are either correlated in $\nu$ but independent in i or vice versa). We show that the standard Hopfield model of neural networks with N neurons can store $N/(\gamma \log N)$ or $\alpha N$ correlated patterns (depending on which notion of storage is used), provided that the correlation comes from a homogeneous Markov chain. This answers the open question whether the standard Hopfield model can store any increasing number of correlated patterns at all in the affirmative. While our bound on the critical value for $\alpha$ decreases with large correlations, the critical $\gamma$ behaves differently for the different types of correlations.

We consider the following problem. A deck of $2n$ cards labeled consecutively from 1 on top to $2n$ on bottom is face down on the table. The deck is given k dovetail shuffles and placed back on the table, face down. A guesser tries to guess at the cards one at a time, starting from top. The identity of the card guessed at is not revealed, nor is the guesser told whether a particular guess was correct or not. The goal is to maximize the number of correct guesses. We show that, for $k \geq 2 \log_2 (2n) + 1$, the best strategy is to guess card 1 for the first half of the deck and card $2n$ for the second half. This result can be interpreted as indicating that it suffices to perform the order of $\log_2(2n)$ shuffles to obtain a well-mixed deck, a fact proved by Bayer and Diaconis. We also show that if $k = c \log_2 (2n)$ with $1 < c < 2$, then the above guessing strategy is not the best.

We analyze Gittins' Markovian model, as generalized by Varaiya, Walrand and Buyukkoc, in discrete and continuous time. The approach resembles Weber's modification of Whittle's, within the framework of both multi-parameter processes and excursion theory. It is shown that index-priority strategies are optimal, in concert with all the special cases that have been treated previously.

We consider a Gibbs sampler applied to the uniform distribution on a bounded region $R \subseteq \mathbf{R}^d$. We show that the convergence properties of the Gibbs sampler depend greatly on the smoothness of the boundary of R. Indeed, for sufficiently smooth boundaries the sampler is uniformly ergodic, while for jagged boundaries the sampler could fail to even be geometrically ergodic.

We consider nonreversible exchange dynamics in $Z^d$ and prove that the stationary, translation invariant measures satisfy the following property: if one of them is a Gibbs measure with a summable potential ${J_R, R \subset Z^d}$, then all of them are convex combinations of Gibbs measures with the same potential, but different chemical potentials $J_{\{0\}}$.

We study an investment decision problem for an investor who has available a risk-free asset (such as a bank account) and a chosen risky asset. It is assumed that the interest rate for the risk-free asset is zero. The amount invested in the risky asset is given by an Itô process with infinitesimal parameters $\mu (\cdot)$ and $\sigma (\cdot)$, which come from a control set. This control set depends on the investor's wealth in the risky asset. The wealth can be transferred between the two assets and there are charges on all transactions equal to a fixed percentage of the amount transacted. The investor's financial goal is to achieve a total wealth of $a > 0$. The objective is to find an optimal strategy to maximize the probability of reaching a total wealth a before bankruptcy. Under certain conditions on the control sets, an optimal strategy is found that consists of an optimal choice of a risky asset and an optimal choice for the allocation of wealth (buying and selling policies) between the two assets.

We consider a simple stochastic discrete-time epidemic model in a large closed homogeneous population that is not necessarily homogeneously mixing. Rather, each individual has a fixed circle of acquaintances and the epidemic spreads along this social network. In case the number of initially infective individuals stays small, a branching process approximation for the number of infectives is in force. Moreover, we provide a deterministic approximation of the bivariate process of susceptible and infective individuals, valid when the number of initially infective individuals is large. These results are used in order to derive the basic reproduction number and the asymptotic final epidemic size of the process. The model is described in the framework of random graphs.