This is an expository survey of recent developments in the field of rates of convergence and asymptotic expansions in the context of the multi-dimensional central limit theorem. A number of applications are discussed. One of them deals with normal approximations and expansions of distribution functions of a class of statistics which includes functions of sample moments.

Speeds of convergence to normality for sums of independent and identically distributed random vectors in $\mathbb{R}^k, k \geqq 1$, are investigated using the method of operators. Results obtained improve and extend existing results on speeds of convergence for the expectations of both bounded and certain unbounded Borel measurable functions, and nonuniform convergence rates.

Let $S_n$ be a stochastic process with either discrete or continuous time parameter and stationary independent increments. Let $N$ be a stopping time for the process such that $EN < \infty$. If the upper tail of the process distribution, $F$, is regularly varying, certain conditions on the lower tail of $F$ and on the tail of the distribution of $N$ imply that $\lim_{y\rightarrow\infty}P(S_N > y)/(1 - F(y)) = EN$. A similar asymptotic relation is obtained for $\sup_n S_{n \wedge N}$, if $n$ is discrete. These asymptotic results are related to the Wald moment identities and to moment inequalities of Burkholder. Applications are given for exit times at fixed and square-root boundaries.

Let $X(t), t \in \lbrack 0, 1 \rbrack$ be a real valued stochastic process with absolutely continuous sample paths. Let $M(a, X(t))$ denote the number of times $X(t) = a$ for $t \in (0, 1\rbrack$ and $N(a, X(t))$ the number of times $X(t)$ crosses the level $a$ for $t \in (0, 1\rbrack$. Under certain conditions on the joint density function of $X(t)$ and its derivative $X(t)$, integral expressions are obtained for $E \lbrack \prod^k_{i = 1} N(a_i, X(t))^j_i \rbrack$ for $j_i$ positive integers (similarly with $M$ replacing $N$). Examples of Gaussian processes $X(t)$ are found for which $X(0) \equiv 0, EN(a, X(t)) < \infty, a \neq 0$ but $EN(0, X(t)) = \infty$. Also examples of stationary Gaussian processes are given for which $EN(a, X(t)) < \infty$ for all $a, EN^2(0, X(t)) = \infty$ but $E\rbrack N(0, X(t))N(a, X(t)) \rbrack < \infty$ for $a \neq 0$. These examples are used to describe the clustering of the zeros of a certain class of Gaussian processes.

The lead digit behavior of a large class of arithmetic sequences is determined by using results from the theory of uniform distribution $\operatorname{mod} 1$. Theory for triangular arrays is developed and applied to binomial coefficients. A conjecture of Benford's that the distribution of digits in all places tends to be nearly uniform is verified.

The condenser theorem in classical potential theory is studied within the framework of Markov processes and probabilistic potential theory. The condenser charge is expressed in terms of successive balayages of a capacitary measure.

New proofs are given for results by Matthes concerning a certain insensitivity phenomenon displayed, for instance, by many well-known stochastic models from the areas of reliability and telephone engineering.

In this paper, growth of branching processes in random environment is considered. In particular it is shown that this process either "explodes" at an exponential rate or else becomes extinct w.p.1. A classification theorem outlining the cases of "explosion or extinction" is given. To prove these theorems, the associated branching process (the process conditioned on each particle having infinite descent) and the reduced branching process (the particles of the process having infinite descent) are introduced. The method of proof used, in general, is direct probabilistic computation, in contrast with the classical functional iteration method.

We show that (i) Brownian motion conditioned to be positive is Brownian meander; (ii) tied-down Brownian meander is Brownian excursion; and (iii) Brownian bridge conditioned to be positive is Brownian excursion. Using these results we derive the distribution of the suprema of the meander and excursion.

The primary concern of this paper is to calculate the distributions and/or means of the maxima, first entrance times, and occupation times of Brownian meander and Brownian excursion. The method employed is to develop conditioned families of random functions which have Brownian meander (or Brownian excursion) as their weak limit and then use the continuous mapping theorem.

Assuming the existence of fourth moment only, we prove the Berry-Esseen theorem for $U$-statistics. Assuming the third absolute moment, we obtain the order bound $O(n^{-\frac{1}{2}} \log^{\frac{1}{3}}n)$. This improves earlier results of Bickel, and Grams and Serfling.

Every measurable real-valued function $f$ on the space of Wiener process paths $\{W(t): 0 \leqq t \leqq 1\}$ can be represented as an Ito stochastic integral $\int^1_0 \varphi(t, \omega) dW(t, \omega)$ where $\varphi$ is a nonanticipating functional with $\int^1_0 \varphi(t, \omega)^2dt < \infty$ for almost all $\omega$.

We show that a Brownian motion random measure with $q$-dimensional parameter domain almost surely nowhere satisfies a Lipschitz condition of order greater than $\frac{1}{2}$.

Let $X(t)$ be a Gaussian process taking values in $R^d$ and with its parameter in $R^N$. Then if $X_j$ has stationary increments and the function $\sigma^2(t) = E\{|X_j(s + t) - X_j(s)|^2\}$ behaves like $|t|^{2\alpha}$ as $|t| \downarrow 0, 0 < \alpha < 1$, the graph of $X$ has Hausdorff dimension $\min \{N/\alpha, N + d(1 - \alpha)\}$ with probability one. If $X$ is also ergodic and stationary, and if $N - d\alpha \geqq 0$, then the dimension of the level sets of $X$ is a.s. $N - d\alpha$.

Probabilities of events involving the jump of a Markov chain to the state space immediately after the last exit before a given time from a boundary atom are determined, for the most part, by the initial time value of the canonical entrance law corresponding to that atom. Three of these probabilities are calculated in terms of canonical quantities in order to attach a probabilistic meaning to an entrance law decomposition of Reuter's and to improve an analytical condition of Chung's for when the Kolmogorov forward equations are satisfied by the chain's transition matrix.

It is shown that, while the value of the expectation $E(X + Y)$ always depends on the random variables $X$ and $Y$ only through their marginal distributions, the same kind of statement cannot be made for $E(X + Y + Z)$.