In this paper we provide a general framework for the study of the central limit theorem (CLT) for empirical processes indexed by uniformly bounded families of functions $\mathscr{F}$. From this we obtain essentially all known results for the CLT in this case; we improve Dudley's (1982) theorem on entropy with bracketing and Kolcinskii's (1981) CLT under random entropy conditions. One of our main results is that a combinatorial condition together with the existence of the limiting Gaussian process are necessary and sufficient for the CLT for a class of sets (modulo a measurability condition). The case of unbounded $\mathscr{F}$ is also considered; a general CLT as well as necessary and sufficient conditions for the law of large numbers are obtained in this case. The results for empiricals also yield some new CLT's in $C\lbrack 0, 1\rbrack$ and $D\lbrack 0, 1\rbrack$.

This paper is a self-contained survey of most of the results known about oriented percolation. The table of contents below should give an idea of the topics which will be covered. A more detailed account can be found in the first section.

Sharp exponential bounds for the probabilities of deviations of the supremum of a (possibly non-iid) empirical process indexed by a class $\mathscr{F}$ of functions are proved under several kinds of conditions on $\mathscr{F}$. These bounds are used to establish laws of the iterated logarithm for this supremum and to obtain rates of convergence in total variation for empirical processes on the integers.

Let $U$ be the distribution function of the nonnegative passage time of an individual bond of the square lattice, and let $\theta_{0n}$ denote one of the first passage times $a_{0n}, b_{0n}$. We define $N_{0n} = \min\{|r|:r \text{is a route of} \theta_{0n}\},$ where $|r|$ is the number of bonds in $r$. It is proved that if $U(0) > 1/2$ then $\lim_{n \rightarrow \infty} \frac{N^a_{0n}{n}} = \lim_{n \rightarrow \infty} \frac{N^b_{0n}{n}} = \lambda \text{a.s. and in} L^1,$ where $\lambda$ is a constant which only depends on $U(0)$.

We study solutions to stochastic differential equations in Hilbert space. In particular we give sufficient conditions for nonexplosion and for the associated semigroup to be of Feller type. We also give applications to systems of stochastic differential equations.

A renewal theory is developed for sums of independent random variables whose distributions are determined by the current state of a Markov chain (also known as "Markov additive" processes, or "semi-Markov" processes). This theory departs from existing theories in that its conclusions are required to be valid conditionally for a given realization of the Markov Chain. It rests on a peculiar coupling construction which differs markedly from existing coupling arguments.

We develop a strong approximation approach to extended multidimensional renewal theory. The consequences of this approximation are a Bahadur-Kiefer type representation of the renewal process in terms of partial sums, Strassen and Chung type laws of the iterated logarithm. We also give a characterization of the renewal process by four classes of deterministic curves in the sense of Revesz (1982). We generalize our results to the case of non-independent and/or nonidentically distributed random vectors.

Asymptotic normality is proven for spectral density estimates assuming strong mixing and a limited number of moment conditions for the process analyzed. The result holds for a large class of processes that are not linear and does not require the existence of all moments.

Let $X_1, X_2, \cdots$ be an i.i.d. sequence of random variables with a continuous density $f$. We consider in this paper the strong limiting behavior as $n \rightarrow \infty$ of the $k$th largest spacing $M^{(n)}_k$ induced by $X_1, \cdots, X_n$ in the sample range. In the case where $f$ is bounded away from zero inside a bounded interval and vanishes outside, we characterize the limiting behaviour of $M^{(n)}_k$ in terms of the local behavior of $f$ in the neighborhood of the point where it reaches its minimum. In the case where the support of $f$ is an unbounded interval, we prove that for any $k \geq 1, M^{(n)}_k \rightarrow 0$ a.s. as $n \rightarrow \infty$ if and only if the distribution of $X_1$ has strongly stable extremes.

The elements of an arbitrary max-stable sequence are exhibited as functionals of a 2-dimensional Poisson point process. The result is extended to a continuous time max-stable process that is continuous in probability. We define an analogue of a stochastic integral appropriate for this context.

Without using the prime number theorem, we obtain the asymptotics of the $r$th largest prime divisor of a harmonically distributed random positive integer $N$; harmonic asymptotics are obtained from asymptotics of the zeta distribution via Tauberian methods. (Knuth and Trabb-Pardo need a strong form of the prime number theorem to obtain the distributions when $N$ is uniformly distributed.) A trick brings in Poisson variates, and then we can use the methods developed for the fractional length of the $r$th longest cycle in a random permutation.

Let $X_i \geq 0$ be independent, $i = 1, \cdots, n$, and $X^\ast_n = \max(X_1, \cdots, X_n)$. Let $t(c) (s(c))$ be the threshold stopping rule for $X_1, \cdots, X_n$, defined by $t(c) = \text{smallest} i$ for which $X_i \geq c(s(c) = \text{smallest} i$ for which $X_i > c), = n$ otherwise. Let $m$ be a median of the distribution of $X^\ast_n$. It is shown that for every $n$ and $\underline{X}$ either $EX^\ast_n \leq 2EX_{t(m)}$ or $EX^\ast_n \leq 2EX_{s(m)}$. This improves previously known results, [1], [4]. Some results for i.i.d. $X_i$ are also included.

The problem of the elliptical symmetry of an operator-stable measure on a finite dimensional vector space was studied by J. P. Holmes, W. N. Hudson and J. D. Mason (1982). The aim of this paper is to consider an analogous question for operator semi-stable measures. We prove a theorem characterizing an elliptically symmetric full operator semi-stable measure. After such a generalization some results for operator-stable measures are obtained as corollaries. At the same time, the methods of proofs seem to be simpler than those in [1], in particular, the theory of Lie algebras is not involved. We also give a description of a full operator-stable and operator semi-stable measure $\mu$ in terms of its quasi-decomposability group, namely the group $\mathbb{G}(\mu) = \{t > 0: \exists A \in \operatorname{Aut} V, h \in V \quad\text{such that}\quad\mu^t = A\mu \ast \delta(h)\}.$

A transition kernel $\mu = (\mu_y)_{y\in Y}$ between Polish spaces $X$ and $Y$ is completely orthogonal if there is a perfect statistic $\varphi: X \rightarrow Y$ for $\mu$, i.e. the fibers of the Borel map $\varphi$ separate the $\mu_y$. Equivalent properties are: a) orthogonal, finitely additive measures $p, q$ on $Y$ induce orthogonal mixtures $\mu^p, \mu^q$ on $X$; b) the Markov operator defined by $\mu$ is subjective on a certain class of Borel functions.