This paper proves large deviation theorems for a general class of random vectors taking values in $\mathbb{R}^d$ and in certain infinite dimensional spaces. The proofs are based on convexity methods. As an application, we give a new proof of the large deviation property of the empirical measures of finite state Markov chains (originally proved by M. Donsker and S. Varadhan). We also discuss a new notion of stochastic convergence, called exponential convergence, which is closely related to the large deviation results.

Let $\{X_j: \mathbf{j} \in J^d\}$ be an array of independent random variables, where $J^d$ denotes the $d$-dimensional positive integer lattice. The main purpose of this paper is to obtain a functional law of the iterated logarithm (LIL) for suitably normalized and smoothed versions of the partial-sum process $S(B) = \sum_{j \in B}X_j$. The method of proof involves the definition of a set-indexed Brownian process, and the embedding of the partial-sum process in this Brownian process. In addition, the LIL is derived for this Brownian process. The method is extended to yield a uniform central limit theorem for the partial-sum process.

Sufficient conditions for the validity of the upper and lower inequality of the law of iterated logarithm for weighted sums $\sum^n_{k = 1} a_{nk}X_k$ of i.i.d. random variables $(X_i)$ are given.

Let $\{X_k\}$ be a sequence of independent, identically distributed, nondegenerate random variables and $S_n = X_1 + \cdots + X_n$. Define $G(x) = P\{|X| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF(y), Q(x) = G(x) + K(x)$ for $x > 0$, and $\{a_n\}$ by $Q(a_n) = n^{-1}$ for large $n$. Let (A) denote the condition: $\lim \sup_{x \rightarrow \infty} G(x)/K(x) < \infty$. We show that (A) implies the following: there exist $\varepsilon > 0, C > 0$, such that for each $M > 0$ a sequence $\{b_n\}$ and a positive constant $c$ can be found for which $c \leqq a_nP\{S_n \in (x - \varepsilon, x + \varepsilon)\} \leq C$ whenever $|x - b_n| \leq Ma_n$ and $n$ is sufficiently large. In fact, the upper bound is valid for all $x$. We also prove that (A) is necessary for either the upper bound result or the lower bound result so that these results are equivalent. Feller had shown that (A) is equivalent to the existence of $\{\gamma_n\}, \{\delta_n\}$ such that the sequence $\{(S_n - \delta_n)/\gamma_n\}$ is stochastically compact.

Let $(S_j)$ be a lattice random walk, i.e. $S_j = X_1 + \cdots + X_j$ where $X_1, X_2, \cdots$ are independent random variables with values in the integer lattice and common nondegenerate distribution $F$, and let $L_n(x) = \sum^{n - 1}_{j = 0} 1_{\{x\}} (S_j)$, the local time of the random walk at $x$ before time $n$. Define $G(x) = P\{|X_1| > x\}, K(x) = x^{-2} \int_{|y| \leq x}y^2 dF (y), Q(x) = G(x) + K(x)$ for $x > 0. Q$ is continuous, strictly decreasing for large $x$, and tends to zero. Thus $a_y$ may be defined by $Q(a_y) = y^{-1}$ for large $y$ and then we let $c_n = a_{n/\log\log n}$. The basic assumption is that $\lim \sup_{x \rightarrow \infty} G(x)/K(x) < 1$ and $EX_1 = 0$. We prove that there exist positive constants $\theta_1, \theta_2$ such that $\lim \sup_{n \rightarrow \infty} c_nn^{-1}L_n(x) = \theta_1$ a.s. for all $x, \lim \sup_{n \rightarrow \infty} \sup_xc_nn^{-1}L_n(x) = \theta_2$ a.s. Furthermore $\lim_{\delta \rightarrow 0} \lim \sup_{n \rightarrow \infty} \sup_{|x - y| \leq \delta c_n}c_nn^{-1}|L_n(x) - L_n(y)| = 0 a.s.$ One of the main tools is an estimate for the "absolute potential kernel": $\sum^\infty_{n = 0}| P\{S_n = z\} - P\{S_n = z + x\}| \leq C(|x|Q(|x|))^{-1}$ assuming strong aperiodicity.

We envision a network of $N$ paths in the plane determined by $N$ independent, two-dimensional Brownian motions $W_i(t), t \geq 0, i = 1, 2, \cdots, N$. Our problem is to study the set of "confluences" $z$ in $\mathbb{R}^2$ where all $N$ paths meet and also the set $M_0$ of $N$-tuples of times $\mathbf{t} = (t_1, \cdots, t_N)$ at which confluences occur: $M_0 = \{\mathbf{t}: W_1(t_1) = \cdots = W_N(t_N)\}$. The random set $M_0$ is analyzed by constructing a convenient stochastic process $X$, which we call "confluent Brownian motion", for which $M_0 = X^{-1}(0)$ and using the theory of occupation densities. The problem of confluences is closely related to that of multiple points for a single process. Some of our work is motivated by Symanzik's use of Brownian local time in quantum field theory.

We show how to use local times to analyze the self-intersections of random fields. In particular, we compute the Hausdorff dimension of $r$-multiple times for Brownian motion in the plane, Brownian sheets and Levy's multiparameter Brownian motion.

Strong regularity for stationary discrete random fields is discussed. An extension of the classical Beurling's Theorem to functions of several variables is given. Necessary and sufficient conditions for the moving average representation of stationary random fields are obtained. A recipe formula for the best linear extrapolator is also given.

We prove that bounded multiparameter martingales converge almost surely if the underlying $\sigma$-fields are generated by a Markov random field which satisfies Dobrushin's uniqueness condition. An example shows that it is not enough to assume that the Markov field is uniquely determined by its conditional probabilities.

Let $(X_n)_{n \in \mathscr{X J}}$ be a sequence of r.v.'s with $E X_n = 0, E(\sum^n_{i = 1} X_i)^2/n \rightarrow \sigma^2 > 0, \sup_{n,m}E(\sum^{m + n}_{i = m + 1} X_i)^2/n < \infty$. We prove the functional c.l.t. for $(X_n)$ under assumptions on $\alpha_n(k) = \sup\{|P(A \cap B) - P(A)P(B)|:A \in \sigma(X_i: 1 \leq i \leq m), B \in \sigma(X_i: m + k \leq i \leq n), 1 \leq m \leq n - k\}$ and the asymptotic behaviour of $\|X_n\|_\beta$ for some $\beta \in (2, \infty\rbrack$. For the special cases of strongly mixing sequences $(X_n)$ with $\alpha(k) = \sup \alpha_n(k) = O(k^{-a})$ for some $a > 1$, or $\alpha(k) = O(b^{-k})$ for some $b > 1$, we obtain functions $f_\beta(n)$ such that $\|X_n\|_\beta = o(f_\beta(n))$ for some $\beta \in (2, \infty\rbrack$ is sufficient for the functional c.l.t., but the c.l.t. may fail to hold if $\|X_n\|_\beta = O(f_\beta(n))$.

Let $\bar{X}$ be the mean of a random sample from a distribution which is symmetric about its unknown mean $\mu$ and has known variance $\sigma^2$. The classical method of constructing a hypothesis test or confidence interval for $\mu$ is to use the normal approximation to $n^{\frac{1}{2}}(\bar{X} - \mu)/\sigma$. In order to make this procedure more robust, we might lightly trim the mean by removing extremes from the sample. It is shown that this procedure can greatly improve the rate of convergence in the central limit theorem, but only if the new mean is rescaled in a rather complicated way. From a practical point of view, the removal of extreme values does not make the test or confidence interval more robust.

We discuss the problem of allocating effort among several competing projects, the states of which evolve according to one-dimensional diffusion processes. It is shown that the "play-the-leader" policy of continuing the project with the leading Gittins index is optimal, and very explicit computations of the index are offered. The question of constructing the diffusions according to the above policy is also addressed.

For a random sequence of events, with indicator variables $X_i$, the behavior of the expectation $E\{(X_k + \cdots + X_{k + m - 1})/(X_1 + \cdots + X_n)\}$ for $1 \leq k \leq k + m - 1 \leq n$ can be taken as a measure of clustering of the events. When the measure on the $X$'s is i.i.d., or even exchangeable, a symmetry argument shows that the expectation can be no more than $m/n$. When the $X$'s are constrained only to be a stationary sequence, the bound deteriorates, and depends on $k$ as well. When $m/n$ is small, the bound is roughly $2m/n$ for $k$ near $n/2$ and is like $(m/n) \log n$ for $k$ near 1 or $n$. The proof given is partly constructive, so these bounds are nearly achieved, even though there is room for improvement for other values of $k$.

A simple proof is given of the Thouvenot relative isomorphism theorem for conditional very weak Bernoulli processes. As a special case, one obtains a simple proof of the Ornstein isomorphism theorem for very weak Bernoulli processes.

We give new algorithms for simulating a flip of an unbiased coin by flipping a coin of unknown bias. We are interested in efficient algorithms, where the expected number of flips is our measure of efficiency. Other authors have represented algorithms as lattices, but by representing them instead as trees we are able to produce an algorithm more efficient than any previously appearing. We also prove a conjecture of Hoeffding and Simons that there is no optimal algorithm. Further, we consider generalizations where the input is a sequence of iid discrete random variables and the output is a uniform random variable with $N$ possible outcomes. In this setting we provide an algorithm significantly superior to those previously published.

Markov exchangeability, a generalization of exchangeability that was proposed by de Finetti, requires that a probability on a string of letters be constant on all strings which have the same initial letter and the same transition counts. The set of Markov exchangeable measures forms a convex set. A graph theoretic and an urn interpretation of the extreme points of this convex set is given.

The central result of this paper consists in proving that all stable densities are bell-shaped (i.e. its $k$th derivative has exactly $k$ zeros and they are simple) thereby generalizing the well-known property of the normal distribution and the associated Hermite polynomials.

Sufficient conditions are developed for various versions of the weighted empirical quantile process to converge weakly in $L^2(0, 1)$ to a weighted Brownian bridge. The results are directly applicable to the derivation of the asymptotic distribution of goodness of fit tests based on the sample quantiles that can be written as a functional defined on $L^2(0, 1)$ continuous in the norm topology. In the process, tight bounds for the moments of transformed uniform order statistics are derived that are likely to have applications elsewhere.

Let $X_i \geq 0$ be i.i.d. random variables with $E(X_i) = \infty$. Then for suitable functions $\varphi$ we have $\overline{\varphi(X)}/\varphi(\bar{X}) \rightarrow 0$ a.s. We give some applications of this result.

Let $P$ be a Dirichlet process with parameter $\alpha$ on $(R, B)$, where $R$ is the real line, $B$ is the $\sigma$-field of Borel subsets of $R$ and $\alpha$ is a non-null finite measure on $(R, B)$. By the use of characteristic functions we show that if $Q(\cdot) = \alpha(\cdot)/\alpha(R)$ is a Cauchy distribution then the mean $\int_R x dP(x)$ has the same Cauchy distribution and that if $Q$ is normal then the distribution of the mean can be roughly approximated by a normal distribution. If the 1st moment of $Q$ exists, then the distribution of the mean is different from $Q$ except for a degenerate case. Similar results hold also in the multivariate case.

Let $J = \{1, 2, \cdots\}^d$ and let $\{X_j, \mathbf{j} \in J\}$ be iid with finite mean. Let $S(nA)$ be the sum of those $X_j$'s for which $\mathbf{j}/n \in A$. It is proved in this paper that $S(\cdot)$ satisfies a strong law of large numbers that is uniform over $A \in \mathscr{A}$, where $\mathscr{A}$ is a family of subsets of $\lbrack 0, 1\rbrack^d$ satisfying a mild condition.

A theorem of Harris states that a monotone Markov process on a finite partially ordered set has positive correlations at time $t$ (assuming positive correlations at time 0) if and only if each jump of the process is either up or down. A new proof of the sufficiency of the jump condition is presented.

We consider random walks on lattices with finite memory and a finite number of possible steps. Using a local limit theorem, we generalize Polya's theorem to such walks, describe how to compute tail probabilities when the number of steps is large, and obtain asymptotic estimates for the average number of points visited.

Let $S_n = X_1 + \cdots + X_n$ where $(X_n)$ is a sequence of 0-mean i.i.d. random vectors in a $B$-space such that $P(\|X_n\| > t) \leq CP(|X_0| > t)$ for some random variable $X_0 \in L_p$. We show that $S_n/n^{1/p} \rightarrow 0$ in $L_p$ iff $B$ is $p$-stable $(1 \leq p < 2)$.