Let $X(t), -\infty < t < \infty$, be a real stationary stochastic process with continuous sample functions. For $t > 0$, put $L_t(u) =$ Lebesgue measure of $\{s: 0 \leq s \leq t, X(s) > u\}$ and $M(t) = \max(X(s): 0 \leq s \leq t)$. For several years the author has studied the limiting properties of these random variables in the case where $X(t)$ is a Gaussian process and under two kinds of limiting operations: i) $t$ fixed and $u \rightarrow \infty$; ii) $t \rightarrow \infty$ and $u = u(t) \rightarrow \infty$ as a function of $t$. The purpose of this paper is to show how the methods developed in the Gaussian case can be extended to the general, not necessarily Gaussian case. This is illustrated by applications of some of the results to specific examples of non-Gaussian processes, and classes of processes containing a Gaussian subclass.

For a new class of Gaussian function space integrals depending upon $n \in \{1, 2,\cdots\}$, the exponential rate of growth or decay as $n \rightarrow \infty$ is determined. The result is applied to the calculation of the specific free energy in a model in statistical mechanics. The physical discussion is self-contained. The paper ends by proving upper bounds on certain probabilities. These bounds will be used in a sequel to this paper, in which asymptotic expansions and limit theorems will be proved for the Gaussian integrals considered here.

We study attractive nearest neighbor spin systems on the integers having "all 0's" and "all 1's" as traps. A necessary and sufficient condition for stability under small one-sided perturbations of the flip rates is established.

In this paper we study the local behavior of empirical processes for independent identically distributed random variables on the real line. The results are applied to get best rates of convergence for various types of density estimators as well as error estimates for the Bahadur representation of the quantile process obtained by Kiefer.

A generalization to the case of independent but not necessarily identically distributed two-dimensional underlying random vectors is obtained of results on univariate empirical df's of van Zuijlen (1976) (linear bounds), Ghosh (1972) and Ruymgaart and van Zuijlen (1978b). No conditions are imposed on the dependence structure of the underlying df's. In the process improvements of van Zuijlen's results concerning linear bounds in the univariate non-i.i.d. case are obtained, whereas also applications of the results on multivariate empirical df's are discussed. Extensions of the two-dimensional results to the $k$-dimensional case $(k > 2)$ are straightforward and therefore omitted.

A general theorem of Hoeffding (Hoeffding (1948)) implies that the limit distribution of symmetric statistics under natural conditions is normal, but rather frequently the variance of this normal distribution vanishes. Even in the very simple case when the statistics are elementary symmetric polynomials of independent random variables $\{X_i: P(X_i = \pm 1) = \frac{1}{2}, i = 1,2, \cdots, n\}$, the limit distributions (in general) are not normal. These distributions will be determined below.

The probability of the event $|S \cap T| = \infty$ is investigated, where $S$ is the trace of a random walk on the set of positive integers and $T$ is a fixed set of natural numbers.

A classical result due to Daniels is that the strength of a bundle of parallel fibres is asymptotically normally distributed. Extensions of this result are obtained and applied to a series-parallel model consisting of a long chain of bundles arranged in series. This model is of importance in studying the reliability of fibrous materials. Improved approximations are also obtained which reduce the error associated with Daniels' approximation both for the single bundle and for the series-parallel system.

For an array $\{X_{ni}\}$ of independent, uniformly null random variables, several necessary and sufficient conditions are given for the convergence in distribution of its extremal process $\mathbf{M}_n = (M^1_n, M^2_n, \cdots)$ as $n \rightarrow \infty$, where $M^k_n(t) = k$th largest $\{X_{ni}: i/n \leq t\}, t > 0$. It is shown that if $\mathbf{M}_n$ converges, then its limit is an extremal process of a Poisson process on the plane. The limit cannot be an extremal process of a non-Poisson, infinitely divisible point process, which is possible for certain stationary variables. A characterization of the convergence of $\mathbf{M}_n$, without the uniformly null assumption, is also given.

We study a recursive algorithm which includes the multidimensional Robbins-Monro and Kiefer-Wolfowitz processes. The assumptions on the disturbances are weaker than the usual assumption that they be a martingale difference sequence. It is shown that the algorithm can be represented as a weighted average of the disturbances. This representation can be used to prove asymptotic results for stochastic approximation procedures. As an example, we approximate the one-dimensional Kiefer-Wolfowitz process almost surely by Brownian motion and as a byproduct obtain a law of the iterated logarithm.

For stationary channels, channel entropy is defined as the supremum of the conditional output entropy over all stationary input sources. If the channel is $\bar{d}$-continuous and conditionally almost block independent, channel entropy gives an upper bound to the output rate of the channel. Furthermore, the channel can be approximated arbitrarily well in the $\bar{d}$-metric by any primitive channel whose noise source entropy exceeds channel entropy and cannot be so approximated if the noise source entropy is less than channel entropy.

Multi-parameter martingales and amarts can be studied using methods developed for amarts defined on a directed set by A. Millet and L. Sucheston. To study an amart indexed by $\mathbb{N} \times \mathbb{N}$, we use an associated process indexed by the "lower layers" of $\mathbb{N} \times \mathbb{N}$. J. B. Walsh's convergence theorem for two-parameter strong martingales is recovered as a special case. Vector-valued versions of some of the results are also stated.

All first hitting times for a one-dimensional diffusion belong to the Bondesson class of infinitely divisible distributions on $\lbrack 0, \infty\rbrack$. A distribution in this class can be conveniently represented in terms of its canonical measure. In this paper we establish a link between the canonical measure of a hitting time and the spectral measure of the differential generator of the diffusion. In particular, it is shown that the derivative of the canonical measure with respect to natural scale (as a function of the point being hit) equals the spectral measure of the differential generator on a restricted interval. The canonical measure is then calculated for several examples arising from the Bessel diffusion process, including the inverse of a gamma variate and the Hartman-Watson mixing distribution.

Let $B_t$ be a $d$-dimensional Brownian motion starting at zero and $h(t)$ a positive nondecreasing function of $t > 0$. It is shown that $\lim \sup_{s\downarrow 0}(1/s)\operatorname{meas} \{t \in (0, s\rbrack: |B_t| > h(t)\sqrt t\} = 1 - e^{-4(q-1)}$; where $q$ is defined as a simple functional of $h$ and, if we take $h_c(t) = c\sqrt{(2 \log \log(1/t))}$ for $h$ and if $0 < c \leqq 1, q = 1/c^2$. We also investigate $X^\ast = \lim \sup_{s\downarrow 0}(1/s)\operatorname{meas}\{t \in (0, s\rbrack: |B_t| < (\sqrt t)/h(t)\}$ and find upper and lower bounds of $X^\ast$, which indicate in particular that if $h = h_c(c > 0), X^\ast = p_c$ (say) is positive and less than one and tends to zero (one) as $c\uparrow \infty$ (respectively, $c \downarrow 0$). The problem for the case $s \rightarrow \infty$ is also treated.

Let $\tilde{W} = \tilde{W}_t, 0 \leq t \leq 1$, be the pinned Wiener process and let $\xi = \int^1_0|\tilde{W}|$. We show that the Laplace transform of $\xi, \phi(s) = Ee^{-\xi s}$ satisfies \begin{equation*}\tag{*}\int^\infty_0 e^{-us}\phi(\sqrt 2 s^{3/2})s^{-1/2} ds = - \sqrt \pi Ai(u)/Ai'(u)\end{equation*} where $Ai$ is Airy's function. Using $(\ast)$, we find a simple recurrence for the moments, $E\xi^n$ (which seem to be difficult to calculate by direct or by other techniques) namely $E\xi^n = e_n \sqrt \pi(36 \sqrt 2)^{-n}/\Gamma \big(\frac{3n + 1}{2}\big)$ where $e_0 = 1, g_k = \Gamma(3k + \frac{1}{2})/\Gamma(k + \frac{1}{2})$ and for $n \geq 1$, $e_n = g_n + \sum^n_{k=1} e_{n-k}\binom{n}{k} \frac{6k + 1}{6k - 1} g_k.$

L. A. Shepp [1] has studied the distribution of the integral of the absolute value of the pinned Wiener process, and has expressed the moment generating function in terms of a Laplace transform. Here we apply Shepp's results to obtain an integral for the density of the distribution. This integral is then evaluated by numerical integration along a path in the complex plane.

Let $W(t)$ be a two dimensional Brownian motion with $W(0) = (1, 0)$ and let $\varphi(t)$ be the net number of times the path has wound around (0, 0), counting clockwise loops as $-1$, counterclockwise as $+1$. Spitzer has shown that as $t \rightarrow \infty, 4\pi\varphi(t)/\log t$ converges to a Cauchy distribution with parameter 1. In this paper we will use Levy's result on the conformal invariance of Brownian motion to give a simple proof of Spitzer's theorem.

A conjecture in geometric probability about the asymptotic normality of the $r$-content of the $r$-simplex, whose $r + 1$ vertices are independently uniformly distributed random points of which $p$ are in the interior and $r + 1 - p$ are on the boundary of a unit $n$-ball, is proved by Ruben (1977). In this article it is shown that the exact density of the random $r$-content is available in the most general case. The technique of inverse Mellin transform is used to get the exact density, thus requiring the knowledge of the $k$th moment of the $r$-content for all real $k$. This $k$th moment is already available in the literature. Approximations and asymptotic results as well as a simpler alternate proof for the conjecture are also given.

The age distribution for the supercritical age dependent branching processes is shown to converge on the set of nonextinction to a particular distribution function if the offspring distribution $\{p_j\}$ satisfies $1 < \Sigma jp_j < \infty$.

A class of multivariate distributions with new better than used (NBU) marginals is introduced. A number of necessary and sufficient conditions for a distribution to be a member in the class are given. Closure results, which are useful for the identification or construction of members in the class, are derived. In particular, simple proofs of some well known preservation properties of the class of the univariate NBU distributions are obtained. Two examples of replacement models, that give rise to multivariate NBU distributions in the class, are discussed.

Consider three basic operations arising in reliability theory: finite mixtures, finite convolutions, formation of coherent systems of independent components. In this paper it is shown that: (i) The class of all phase type distributions is closed under all three operations; (ii) the class of all phase type distributions having a representation in which the matrix is upper triangular is the smallest class which is closed under all three operations and contains all exponential distributions. The paper includes a section of preliminaries in which the relevant material regarding phase type distributions and reliability theory is summarized.

It has been conjectured, for any discrete density function $\{p_j\}$ on the integers, that there exists an $n_0$ such that the $n$-fold convolution $\{p_j\}^{\ast n}$ is unimodal for all $n \geq n_0$. A similar conjecture has been stated for continuous densities. We present several counterexamples to both of these conjectures. As a positive result, it is shown for a discrete density with a connected 3-point integer support that its $n$-fold convolution is fully unimodal for all sufficiently large $n$.