This is a survey article about occupation densities for both random and nonrandom vector fields $X: T \rightarrow \mathbb{R}^d$ where $T \subset \mathbb{R}^N$. For $N = d = 1$ this has previously been called the "local time" of $X$, and, in general, it is the Lebesgue density $\alpha(x)$ of the occupation measure $\mu(\Gamma) =$ Lebesgue measure $\{t\in T: X(t)\in \Gamma\}$. If we restrict $X$ to a subset $A$ of $T$ we get a corresponding density $\alpha(x, A)$ and we will be interested in its behavior both in the space variable $x$ and the set variable $A$. The first part of the paper deals entirely with nonrandom, nondifferentiable vector fields, focusing on the connection between the smoothness of the occupation density and the level sets and local growth of $X$. The other two parts are concerned, respectively, with Markov processes $(N = 1)$ and Gaussian random fields. Here the emphasis is on the interplay between the probabilistic and real-variable aspects of the subject. Special attention is given to Markov local times (in the sense of Blumenthal and Getoor) as occupation densities, and to the role of local nondeterminism in the Gaussian case.

Suppose that the properly normalized partial sums of a sequence of independent identically distributed random variables with values in a separable Banach space converge in distribution to a stable law of index $\alpha$. Then without changing its distribution, one can redefine the sequence on a new probability space such that these partial sums converge in probability and consequently even in $L^p (p < \alpha)$ to the corresponding stable process. This provides a new method to prove functional central limit theorems and related results. A similar theorem holds for stationary $\phi$-mixing sequences of random variables.

An ergodic theorem is proved which extends the subadditive ergodic theorem of Kingman and the Banach valued ergodic theorem of Mourier. The theorem is applied to several problems, in particular to a problem on empirical distribution functions.

An upper bound for $E(\sup_{0\leqslant s\leqslant\tau}\|M_s\|^2)$, where $M$ is a square integrable martingale and $\tau$ a stopping time is given in terms of $\lbrack M\rbrack_{\tau^-}$ and $\langle M\rangle_{\tau^-}$. Counter examples show that $4E(\langle M\rangle_{\tau^-})$, which is easily derived as an upper bound from a classical Doob's inequality, when $\tau$ is predictable or totally unaccessible, is no longer an upper bound in general. The obtained majoration is used to prove existence and uniqueness of strong solutions of a stochastic equation $dX_t = a(t, X) dZ_t$, where $a$ is a functional, depending possibly on the whole past of $X$ before $t$, and $Z$ is a semimartingale. Our result thus extends to systems "with memory" recent results by Protter, Kazamaki, Doleans-Dade and Meyer.

Let $Z = (Z_0, Z_1, \cdots)$ be a sequence of random variables taking values in a countable state space $I$. We use a generalization of exchangeability called partial exchangeability. $Z$ is partially exchangeable if for two sequences $\sigma, \tau \in I^{n+1}$ which have the same starting state and the same transition counts, $P(Z_0 = \sigma_0, Z_1 = \sigma_1, \cdots, Z_n = \sigma_n) = P(Z_0 = \tau_0, Z_1 = \tau_1, \cdots, Z_n = \tau_n)$. The main result is that for recurrent processes, $Z$ is a mixture of Markov chains if and only if $Z$ is partially exchangeable.

The problem of coding a stationary process $\{X_i\}^\infty_{i=-\infty}$ onto a stationary process $\{Y_i\}^\infty_{i=-\infty}$ so that for some positive integer $m, (Y_0, Y_1, \cdots, Y_{m-1})$ has a given marginal distribution is considered. The problem is solved for $\{X_i\}$ nonergodic as well as ergodic. The associated universal coding problem is also solved, where one seeks to find a coding function which yields the desired marginal distribution for each member of a class of possible distributions for $\{X_i\}$.

It is shown that, when conditional on a set of given average values, the frequency distribution of a series of independent random variables with a common finite distribution converges in probability to the distribution which has the maximum relative entropy for the given mean values.

Let $N$ balls be distributed independently and at random into $n$ boxes. Let $\rho_{nj}$ denote the number of balls in the $j$th box. Let $(c_0, c_1, c_2, \cdots)$ be a sequence of real numbers. Three limit theorems are proved for the sum $\sum^n_{j=1}c_{\rho_{nj}}$ as $N$ and $n$ tend to infinity in such a way that $N/n \rightarrow 0$.

If $X_1, X_2, \cdots$ are random variables with values in (0, 1), let $D_{n1}, \cdots, D_{n,n+1}$ denote the $n + 1$ spacings given by the first $n$ observations, $X_1, \cdots, X_n$. If $G^\ast_n$ denotes the empirical distribution function of the normalized spacings $\{(n + 1)D_{ni}\}$, it is proved in this paper that under the Kakutani model in which $X_m$ is a uniform random variable over the largest spacing determined by $X_1, \cdots, X_{m-1}$, with probability one $G^\ast_n \rightarrow G$ uniformly, where $G$ is the uniform distribution function on (0, 2). This is in sharp contrast to the known exponential limiting distribution when the $X_i$ are independent uniform random variables on (0, 1).

Let $X_t$ be standard Brownian motion in $R^d$ starting at fixed $X_0$, and let $T$ be the hitting time for a sphere or concentric spherical shell. Explicit formulas, in terms of a natural Laplace-Gegenbauer transform, are obtained for the joint distribution of $T$ and $X_T$.

Let $Y_1, \cdots, Y_r$ be independent random variables, each uniformly distributed on $\mathscr{M} = \{1,2, \cdots, M\}$. It is shown that at most $N = 1 + M + \cdots + M^{r-1}$ pairwise independent random variables, all uniform on $\mathscr{M}$ and all functions of $(Y_1, \cdots, Y_r)$, can be defined. If $M = p^k$ for some prime $p$, the maximum can be attained by a strictly stationary sequence $X_1, \cdots, X_N$, for which any $r$ successive random variables are independent.

Relationships between the growth of a sequence $N_k$ and conditions on the tail of the distribution of a sequence $X_l$ of i.i.d. mean zero random variables are given that are necessary and sufficient for $$\sum^\infty_{k=1} P\big\{\big|\frac{1}{N_k} \sum^{N_k}_{l=1} X_l\big| > \varepsilon\big\} < \infty.$$ The results are significant for distributions satisfying $E(|X_l|) < \infty$ but $E(|X_l|^\beta) = \infty$ for some $\beta > 1$. Necessary and sufficient conditions for the finiteness of sums of the form $$\sum^\infty_{n=1} \gamma (n)P\big\{\big|\frac{1}{n} \sum^n_{l=1} X_l\big| > \varepsilon\big\}$$ are obtained as a corollary.