We consider a sequence $\xi_1, \xi_2, \cdots$ of independent and identically distributed random observations. Let $\mu_n$ denote the empirical distribution for the sample $\{\xi_1, \cdots, \xi_n\}$. It is the aim of the present article to give a survey of various results in the theory of empirical distributions and empirical processes. Special emphasis is given to the developments of the last ten years.

The excursion straddling $t$ and the first excursion exceeding $u$ in length of a Markov process are compared in a general setting. These results are then specialized to the excursions from a point, and to specific processes. The main tool is the general theory of excursions as developed by Maisonneuve and others.

The analogue of Hammersley's theorem on the length of the longest monotonic subsequence of independent, identically, and continuously distributed random variables is obtained for the pseudorandom van der Corput sequence. In this case there is no limit but the precise limits superior and inferior are determined. The constants obtained are closely related to those established in the independent case by Logan and Shepp, and Vershik and Kerov.

Two different stochastic integrals have been developed during the last ten years. One is largely associated with the work of E. J. McShane (the star integral), and the other has grown out of the work of H. Kunita and S. Watanabe (the dot integral). Assuming the customary conditions that guarantee the existence of the star integral, we give a formula relating the two integrals. We show that the star integral is equal to the dot integral provided one takes a projection of the integrand onto the space of predictable processes before evaluating the dot integral. This essentially embeds the theory of the star integral into that of the dot integral.

Let $Y_1, Y_2, \cdots$ be independent and identically distributed random variables with common distribution function $F$ and let $X_n = \max\{Y_1, \cdots, Y_n\}$ for $n = 1, 2, \cdots$. Necessary and sufficient conditions (in terms of $F$) are derived for the existence of a sequence of positive constants $\{a_n\}$ such that the sequence $\{X_n/a_n\}$ is stochastically compact. Moreover, the relation between the stochastic compactness of partial maxima and partial sums of the $Y_n$'s is investigated.

In this paper, an $L_p$ analogue of Anscombe's theorem is shown to hold and is then applied to obtain the variance and other central moments of the first passage time $T_c = \inf\{n \geqslant 1 : S_n > cn^\alpha\}$, where $0 \leqslant \alpha < 1, S_n = X_1 + \cdots + X_n$ and $X_1, X_2, \cdots$ are i.i.d. random variables with $EX_1 > 0$. The variance of $T_c$ in the special case $\alpha = 0$ has been studied by various authors in classical renewal theory, and our approach in this paper provides a simple treatment and a natural extension (to the case of a general $\alpha$) of this classical result. The related problem concerning the asymptotic behavior of $\max_{j\leqslant n}j^{-\alpha}S_j$ is also studied, and in this connection, certain maximal inequalities are obtained and they are applied to prove the corresponding moment convergence results of the theorems of Erdos and Kac, and of Teicher.

Let $E$ be an infinite-dimensional vector space carrying a Gaussian measure $\mu$ with mean 0 and a measurable norm $q$. Let $F(t) := \mu(q \leqslant t)$. By a result of Borell, $F$ is logarithmically concave. But we show that $F'$ may have infinitely many local maxima for norms $q = \sup_n|f_n|/a_n$ where $f_n$ are independent standard normal variables. We also consider Hilbertian norms $q = (\Sigma b_nf^2_n)^{\frac{1}{2}}$ with $b_n > 0, \Sigma b_n < \infty$. Then as $t \downarrow 0$ we can have $F(t) \downarrow 0$ as rapidly as desired, or as slowly as any function which is $o(t^n)$ for all $n$. For $b_n = 1/n^2$ and in a few closely related cases, we find the exact asymptotic behavior of $F$ at 0. For more general $b_n$ we find inequalities bounding $F$ between limits which are not too far apart.

Suppose that $n$ devices are subjected to shocks occurring randomly in time as events in a Poisson process. Upon occurrence of the $i$th shock the devices suffer nonnegative random damages with joint distribution $F_i$. Damages from successive shocks are independent and accumulate additively. Failure of the $j$th device occurs at the time $T_j$ when its accumulated damage first exceeds its breaking threshold $x_j$. If $\tau$ is the life function of a coherent system, then the system life length $\tau(T_1, \cdots, T_n)$ has a distribution with increasing hazard rate average providing that $F_1, F_2, \cdots$ satisfy a multivariate stochastic ordering condition that depends upon $\tau$. If $F_1 = F_2 = \cdots$ and $\bar{H}$ is the joint survival function of $T_1, \cdots, T_n$, then $\lbrack\bar{H}(\alpha\mathbf{t})\rbrack^{1/\alpha}$ is decreasing in $\alpha$ for all $\mathbf{t} \geqslant 0. \bar{H}$ also satisfies a multivariate "new better than used" property. Moreover $T_1, \cdots, T_n$ are associated when $F_1 = F_2 = \cdots$. Examples of specific distributions are given which arise from the shock model, including a new bivariate gamma distribution which reduces to the bivariate exponential distribution of Marshall and Olkin as a special case.

Several conditions are considered that extend to a multivariate setting the univariate concept of an increasing hazard rate average. The relationships between the various conditions are established. In particular it is shown that if for some independent random variables $X_1, \cdots, X_k$ with increasing hazard rate average and some coherent life functions $\tau_1, \cdots, \tau_n$ of order $k, T_i = \tau_i(X_1, \cdots, X_k)$, then the joint survival function $\bar{F}(\mathbf{t}) = P(T_1 > t_1, \cdots, T_n > t_n)$ has the property that $\alpha^{-1} \log \bar{F}(\alpha\mathbf{t})$ is decreasing in $\alpha > 0$ whenever each $t_i \geqslant 0$. Various other properties of the multivariate conditions are given. The conditions can all be stated in terms of inequalities in which equality implies that the one dimensional marginal distributions are exponential. For most of the conditions, the form of the multivariate exponential distributions that satisfy the equality is exhibited.

The Skorokhod representation method of proving martingale invariance principles has received only limited attention. This paper shows it to be a very powerful tool. We use the representation to obtain sufficient conditions for a sequence of random functions to be tight, and to give a direct proof of an invariance principle.

Let $S$ be a countable set, $Q$ a strictly positive matrix on $S \times S, \mathscr{G}(Q)$ the set of one-dimensional Markov random fields taking values in $S$ determined by $Q$. In this paper a short proof of Kesten's sufficient condition for $\mathscr{G}(Q) = \phi$ is presented.

Consider a system that is composed of $n$ components, each of which is operating at some performance level. We suppose that there exists a nondecreasing function $\phi$ such that $\phi(x_1, \cdots, x_n)$ denotes the performance level of the system when the $i$th component's performance level is $x_i, i = 1, \cdots, n$. We allow both $x_i$ and $\phi(x_1, \cdots, x_n)$ to be arbitrary nonnegative numbers and extend many of the important results of the usual binary model to this more general framework. In particular, we obtain a fundamental inequality for $E\lbrack\phi(X_1, \cdots, X_n)\rbrack$ when $\phi$ is binary, which can, among other things, be used to generate a host of inequalities concerning increasing failure rate average distributions including, as a special case, the convolution and system closure theorem. We also define the concept of an increasing failure rate average stochastic process and prove the analog of the closure theorem; and then also do the same for new better than used stochastic processes.