The Poincaré and Cheeger bounds are two useful bounds for the second largest eigenvalue of a reversible Markov chain. Diaconis and Stroock and Jerrum and Sinclair develop versions of these bounds which involve choosing paths. This paper studies these path-related bounds and shows that the Poincaré bound is superior to the Cheeger bound for simple random walk on a tree and random walk on a finite group with any symmetric generating set. This partially resolves a question posed by Diaconis and Stroock.

We consider the problem of sampling according to a distribution with log-concave density F over a convex body $K \subseteq \mathbf{R}^n$. The sampling is done using a biased random walk and we give improved polynomial upper bounds on the time to get a sample point with distribution close to F.

Given $F \subset [0, 1]^2$ and finite, let $\sigma(F)$ denote the length of the minimal Steiner triangulation of points in F. By showing that minimal Steiner triangulations fit into the theory of subadditive and superadditive Euclidean functionals, we prove under a mild regularity condition that $$\lim_{n \to \infty} \sigma(X_1,\dots, X_n)/n^{1/2} = \beta \int_{[0, 1]^2}f(x)^{1/2} dx \c.c.,$$ where $X_1,\dots, X_n$ are i.i.d. random variables with values in $[0, 1]^2$, $\beta$ is a positive constant, f is the density of the absolutely continuous part of the law of $X_1$ , and c.c. denotes complete convergence. This extends the work of Steele. The result extends naturally to dimension three and describes the asymptotics for the probabilistic Plateau functional, thus making progress on a question of Beardwood, Halton and Hammersley. Rates of convergence are also found.

The partial derivatives with respect to time and the fractional Brownian motion of a particular class of stationary processes are defined. Although the fractional Brownian motion is not semimartingale, the bilinear SDE with fractional Brownian motion input is considered and solved. The solution is explicitly given in both the frequency and time domains in the case when the coefficient of the bilinear term is pure imaginary. The stationary Stratonovich solution of the bilinear SDE with white noise input is also considered.

Sufficient conditions are given for existence and uniqueness in Smoluchowski's coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of nonuniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski's equation.

We wish to describe how a chosen node in a network of queues over- loads. The overloaded node may also drive other nodes into overload, but the remaining "super" stable nodes are only driven into a new steady state with stochastically larger queues. We model this network of queues as a Markov additive chain with a boundary. The customers at the "super" stable nodes are described by a Markov chain, while the other nodes are described by an additive chain. We use the existence of a harmonic function h for a Markov additive chain provided by Ney and Nummelin and the asymptotic theory for Markov additive processes to prove asymptotic results on the mean time for a specified additive component to hit a high level l. We give the limiting distribution of the "super" stable nodes at this hitting time. We also give the steady-state distribution of the "super" stable nodes when the specified component equals l. The emphasis here is on sharp asymptotics, not rough asymptotics as in large deviation theory. Moreover, the limiting distributions are for the unscaled process, not for the fluid limit as in large deviation theory.

We establish strong and weak approximations for quadratic forms of weakly and strongly dependent random variables and obtain necessary and sufficient conditions for the weak convergence of weighted functions of quadratic forms. The results are applied to get the asymptotic distributions of some tests which can be used to detect possible changes in the long-memory parameter.

Consider a process $X(\cdot) = {X(t), 0 \leq t < \infty}$ which takes values in the interval $I = (0, 1)$, satisfies a stochastic differential equation $$dX(t) = \beta(t)dt + \sigma(t)dW(t), X(0) = x \epsilon I$$ and, when it reaches an endpoint of the interval I, it is absorbed there. Suppose that the parameters $\beta$ and $\sigma$ are selected by a controller at each instant $t \epsilon [0, \infty)$ from a set depending on the current position. Assume also that the controller selects a stopping time $\tau$ for the process and seeks to maximize $\mathbf{E}u(X(\tau))$, where $u: [0, 1] \to \Re$ is a continuous "reward" function. If $\lambda := \inf{x \epsilon I: u(x) = \max u}$ and $\rho := \sup{x \epsilon I: u(x) = \max u}$, then, to the left of $\lambda$, it is best to maximize the mean-variance ratio $(\beta/\sigma^2)$ or to stop, and to the right of $\rho$, it is best to minimize the ratio $(\beta/\sigma^2)$ or to stop. Between $\lambda$ and $\rho$, it is optimal to follow any policy that will bring the process $X(\cdot)$ to a point of maximum for the function $u(\cdot)$ with probability 1, and then stop.

For a complete graph $K_n$ on n vertices with weighted edges, define the weight of a spanning tree (more generally, spanning forest) as the product of edge weights involved. Define the tree weight (forest weight) of $K_n$ as the total weight of all spanning trees (forests). The uniform edge weight distribution is shown to maximize the tree weight, and an explicit bound on the tree weight is formulated in terms of the overall variance of edge weights as well as the variance of the sum of edge weights over nodes. An application to sparse random graphs leads to a bound for the relative risk of observing a spanning tree in a well-defined neighborhood of the uniform distribution. An analogous result shows that, for each positive integer k, the weight of all forests with k rooted trees is also maximized under the uniform distribution. A key ingredient for the latter result is a formula for the weight of forests of k rooted trees that generalizes Maxwell's rule for spanning trees. Our formulas also enable us to show that the number of trees in a random rooted forest is intrinsically divisible, that is, representable as a sum of n independent binary random variables $\varepsilon_j$, with parameter $(1 + \lambda_j)^{-1}, \lambda$'s being the eigenvalues of the Kirchhoff matrix. This is directly analogous to the properties of the number of blocks in a random set partition, of the size of the random matching set and of the number of leaves in a random tree.

We consider stochastic approximation algorithms with constant step size whose average ordinary differential equation (ODE) is cooperative and irreducible. We show that, under mild conditions on the noise process, invariant measures and empirical occupations measures of the process weakly converge (as the time goes to infinity and the step size goes to zero) toward measures which are supported by stable equilibria of the ODE. These results are applied to analyzing the long-term behavior of a class of learning processes arising in game theory.

This work investigates some spectral characteristics of the errors of optimal linear predictors for weakly stationary random fields. More specifically, for errors of optimal linear predictors, results here explicitly bound the fraction of the variance attributable to some set of frequencies. Such a bound is first obtained for random fields on $\mathbb{R}^d$ observed on the infinite lattice $\deltaJ$ for all J on the d-dimensional integer lattice. If the spectral density exists, then the faster the spectral density tends to 0 at high frequencies, the more quickly this bound tends to 0 as $\delta \downarrow 0$. Under certain conditions on the spectral density, a similar result is given for processes on $\mathbb{R}$ where both observations and predictands are confined to a finite interval and observations may not be evenly spaced. These results provide a powerful tool for studying a problem the author has previously addressed using different methods: the properties of linear predictors calculated under an incorrect spectral density. Specifically, this work gives a number of new rates of convergence to optimality for predictors based on an incorrect spectral density when the ratio of the incorrect to the correct spectral density tends to 1 at high frequencies.