We consider multivariate empirical processes $X_n(t) := \sqrt n (F_n(t) - F(t))$, where $F_n$ is an empirical distribution function based on i.i.d. variables with distribution function $F$ and $t \in \mathbb{R}^k$. For $X_F$ the weak limit of $X_n$, it is shown that $c(F, k)\lambda^{2(k-1)}e^{-2\lambda^2} \leq P\big\{\sup_t X_F(t) > \lambda\big\} \leq C(k)\lambda^{2(k-1)}e^{-2\lambda^2}$ for large $\lambda$ and appropriate constants $c, C$. When $k = 2$ these constants can be identified, thus permitting the development of Kolmogorov--Smirnov tests for bivariate problems. For general $k$ the bound can be used to obtain sharp upper-lower class results for the growth of $\sup_tX_n(t)$ with $n$.

We introduce a new Brownian bridge approximation to weighted empirical and quantile processes with rates in probability. This approximation leads to a number of general invariance theorems for empirical and quantile processes indexed by functions. Improved versions of the Chibisov-O'Reilly theorems, the Eicker-Jaeschke theorems for standardized empirical and quantile processes, the normal convergence criterion, and various other old and new asymptotic results on empirical and quantile processes are presented as consequences of our general theorems. In the process, we provide a new characterization of Erdos-Feller-Kolmogorov-Petrovski upper-class functions for the Brownian motion in an improved form.

We prove general invariance principles for integral functions of the empirical process. As corollaries we derive probabilistic proofs of the sufficiency criteria for a distribution to belong to the domain of attraction of the normal and stable laws with index $0 < \alpha < 2$. In the process we obtain equivalent statements of these criteria in terms of the tail behaviour of the underlying quantile function. We also give a representation of any stable random variable with index $0 < \alpha < 2$ in terms of a linear combination of two independent and identically distributed Poisson integrals. The role of a fixed number of extreme terms is exactly determined.

For every integer $x$, construct a stationary continuous-time Markov process $\gamma(x; t)$, with state space $\{-1, +1\}$ (all processes independent, and having the same distributions). Consider a particle moving at unit speed along the real line, with its direction completely determined by the $\gamma$'s, as follows: if $S_t$ is its position at time $t$, then $S_0 = 0$ and $S_{i + 1} = S_i + \gamma(S_i; i)$ for $i = 0, 1, 2,\cdots$. The increments are not stationary, nor is $S_n$ Markov, yet this process has much in common with the classical random walk, including zero-one laws, a strong law of large numbers, and an invariance principle. The main result of the paper is the proof of the natural conjecture that the process is recurrent if and only if $P\{\gamma(0; 0) = +1\} = \frac{1}{2}$. We also show how the FKG inequality can be used to investigate this process.

Two mathematical models of chemical reactions with diffusion for a single reactant in a one-dimensional volume are compared, namely, the deterministic and the stochastic models. The deterministic model is given by a partial differential equation, the stochastic one by a space-time jump Markov process. By the law of large numbers the consistency of the two models is proved. The deviation of the stochastic model from the deterministic model is estimated by a central limit theorem. This limit is a distribution-valued Gauss-Markov process and can be represented as the mild solution of a certain stochastic partial differential equation.

Let $X_1, X_2,\cdots$ be an i.i.d. sequence of random variables with a continuous density $f$, positive on $(A, B)$, and null otherwise. Under the assumption that $Y_n = \min\{X_1,\cdots, X_n\}$ and $Z_n = \max\{X_1,\cdots, X_n\}$ belong to the domain of attraction of extreme value distributions and that $f(x) \rightarrow 0$ as $x \rightarrow A$ or $x \rightarrow B$, we show that the weak limiting behavior of $Y_n$ and $Z_n$ characterizes completely the weak limiting behavior of the maximal spacing generated by $X_1,\cdots, X_n$ and obtain the corresponding limiting distributions. We study as examples the cases of the normal, Cauchy, and gamma distributions.

The original Erdos-Renyi theorem states that $U_n/(\alpha k) \rightarrow 1$ almost surely for a large class of distributions, where $U_n = \sup_{0\leq i \leq n - k} (S_{i+k} - S_i), S_i = X_1 + \cdots + X_i$ is a partial sum of i.i.d. random variables, $k = \kappa(n) = \lbrack c \log n\rbrack, c > 0$, and $\alpha > 0$ is a number depending only upon $c$ and the distribution of $X_1$. We prove that the $\lim \sup$ and the $\lim inf$ of $(U_n - \alpha k)/\log k$ are almost surely equal to $(2t^\ast)^{-1}$ and $-(2t^\ast)^{-1}$, respectively, where $t^\ast$ is another positive number depending only upon $c$ and the distribution of $X_1$. The same limits are obtained for the random variable $T_n = \sup_{1\leq i \leq n}(S_{i+\kappa(i)} - S_i)$ studied by Shepp.

Let $X_1, X_2,\cdots$ be a sequence of independent identically distributed random vectors and $S_n = X_1 + \cdots + X_n$. Necessary and sufficient conditions are given for there to exist matrices $B_n$ and vectors $\gamma_n$ such that $\{B_n(S_n - \gamma_n)\}$ is stochastically compact, i.e., $\{B_n(S_n - \gamma_n)\}$ is tight and no subsequential limit is degenerate. When this condition holds we are able to obtain precise estimates on the distribution of $S_n$. These results are then specialized to the case where $X_1$ is in the generalized domain of attraction of an operator stable law and a local limit theorem is proved which generalizes the classical local limit theorem where the normalization is done by scalars.

Let $F$ be a probability distribution on $\mathbb{R}$. Then there exist symmetric (about zero) random variables $X$ and $Y$ whose sum has distribution $F$ if and only if $F$ has mean zero or no mean (finite or infinite). Now suppose $F$ is a probability distribution on $\mathbb{R}^n$. There exist spherically symmetric (about the origin) random vectors $\mathbf{X}$ and $\mathbf{Y}$ whose sum $\mathbf{X + Y}$ has distribution $F$ if and only if all the one-dimensional distributions obtained by projecting $F$ onto lines through the origin have either mean zero or no mean.

Strong invariance principles with other of approximation $O(t^{1/2-\kappa})$ are obtained for sequences of dependent random variables. The basic dependence assumptions include various generalizations of martingales such as asymptotic martingales (amarts), semiamarts, and mixingales as well as processes characterized by a condition on the Doleans measure. Provided the partial sum process is uniformly integrable, also martingales in the limit and games fairer with time are included. Sufficient conditions for linear growth of the covariance function of the partial sums are given.

The paper studies in detail the sample paths of Ito-type stochastic integrals with respect to $p$-stable motion $M(t), t \geq 0$. These results, in turn, permit an analysis of the concept of multiple $p$-stable integrals of the form $\int \cdots \int f(t_1,\cdots, t_n)dM(t_1) \cdots dM(t_n)$, and, in particular, a full description of functions of two variables $f(t_1, t_2)$ for which the double stochastic integral $\int \int f(t_1, t_2) dM(t_1) dM(t_2)$ exists.

We investigate the lower tail of $q_r = (\sum^\infty_{i=1} | \alpha_i\theta_i|^r)^{1/r}$ seminorms on $R^\infty$, where $r \geq 1$ and $\theta_i$ are standard $p$-stable real random variables. We prove that for $p < r \leq 2$ we have $P\{q_r \leq t\} \geq \exp\{-ct^{-pr/(r-p)}\}$ in some neighbourhood of 0, where $c$ is a nonnegative constant. If $r \leq p$, then for any positive, increasing function $f$, we can find $q_r$ such that $P\{q_r \leq t\} \leq f(t)$ for $t \leq 1$. We also give a new characterization of Banach spaces of stable type $p$ in terms of the behaviour of $\mu\{\| \cdot \| \leq t\}$ near 0, where $\mu$ is a symmetric and $p$-stable measure.

Suppose that $E$ is a complete separable real metric vector space. It is proved that if $X$ is a symmetric $E$-valued $p$-stable random vector, $0 < p < 2$, and $q$ is a lower semicontinuous, a.s. finite seminorm, then the distribution of $q(X)$ is absolutely continuous apart from a possible jump. If, additionally, $q$ is strictly convex or $0 < p < 1$, then the distribution of $q(X)$ is either absolutely continuous or degenerate at 0. This result settles, in particular, the problem of absolute continuity of the supremum of stable sequences, extending thus Tsirel'son's theorem.

Let $X_n, n \in \mathbb{N}$ be i.i.d. with mean 0 and variance 1. Let $B \in \sigma(X_n: n \in \mathbb{N})$ be a set such that its distances from the $\sigma$-fields $\sigma(X_1,\cdots, X_n)$ are of order $O(1/n(lg n)^{2 + \varepsilon})$ for some $\varepsilon > 0$. We prove that for those $B$ the conditional probabilities $P((1/\sqrt n)\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by a modified Edgeworth expansion up to order $O(1/n)$. An example shows that this is not true any more if the distances of $B$ from $\sigma(X_1,\cdots, X_n)$ are only of order $O(1/n(lg n)^2)$.

An independent stationary process $\{X_i\}^\infty_{i=1}$ in $\mathbb{R}^n$ is perturbed by a sequence of Euclidean motions to obtain a new process $\{Y_i\}^\infty_{i=1}$. Criteria are given for the singularity or equivalence of these processes. When the distribution of the $X$ process has finite Fisher information, the criteria are necessary and sufficient. Moreover, it is proved that it is exactly under the condition of finite Fisher information that the criteria are necessary and sufficient.

A strengthened central limit theorem for densities is established showing monotone convergence in the sense of relative entropy.