An introduction is provided to the mathematical tools and problems of percolation theory. A discussion of Bernoulli percolation models shows the role of graph duality and correlation inequalities in the recent determination of the critical probability in the square, triangular, and hexagonal lattice bond models. An introduction to first passage percolation concentrates on the problems of existence of optimal routes, length of optimal routes, and conditions for convergence of first passage time and reach processes.

On a countable set of sites $S$, the zero range process is constructed when the stochastic matrix $p(x, y)$ determining the one particle motion satisfies a mild assumption. The set of invariant measures for this process is described in the following two cases: a) The system is attractive and $p(x, y)$ is recurrent. b) The system is attractive, $p(x, y)$ corresponds to a simple random walk on the integers and the rate at which particles leave any site is bounded.

Let $T_n$ denote the length of the minimal triangulation of $n$ points chosen independently and uniformly from the unit square. It is proved that $T_n/\sqrt n$ converges almost surely to a positive constant. This settles a conjecture of Gyorgy Turan.

Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for describing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately rescaled, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.

For random walks $s_n, n = 1,2, \cdots$ whose distribution can be imbedded in an exponential family, a method is described for determining the asymptotic behavior as $m \rightarrow \infty$ of $P\{s_n > m c(n/m) \quad\text{for some}\quad n < m\mid s_m = m \mu_0\}, \quad\mu_0 < c(1).$ Applications are given to the distribution of the Smirnov statistic and to modified repeated significance tests.

There are many results as to unimodality of infinitely divisible distributions. But few are known about strong unimodality of infinitely divisible distributions. In this paper, we consider two subclasses of infinitely divisible distributions and give necessary and sufficient conditions for strong unimodality of distributions belonging to such classes.

We characterize the class of linear operators on a finite dimensional inner product space which are the exponents of a full operator-stable law. This answers a question of Paulauskas [6] concerning those operator-stable laws whose characteristic functions are the exponential of quadratic forms. The symmetry group of such laws must be conjugate to the group of all orthogonal transformations on the space.

Let $\{W(t), 0 \leq t < +\infty\}$ be a standard Wiener process and $0 < b_t \leq t$ be a nondecreasing function of $t$. The properties of the process $Y_1(t) = b^{-1/2}_t \sup_{0\leq s \leq t - b_t}(W(s + b_t) - W(s))$ are investigated. One of the results says that $\lim_{t\rightarrow\infty}(Y_1(t) - (2 \log tb^{-1}_t)^{1/2}) = 0$ a.s. if $b_t$ is "much less" than $t$. Analogous properties of similar processes are studied.

It is proved that genuinely infinite dimensional Brownian motions on $\ell^p$ sequence spaces have natural rates of escape, provided the coordinates are independent. An analogous result holds for separable Hilbert space. Computations of Brownian rates of escape and further properties are considered.

The uniform empirical process $U_n$ is considered as a process indexed by intervals. Powerful new exponential bounds are established for the process indexed by both "points" and intervals. These bounds trivialize the proof of the Chibisov-O'Reilly theorem concerning the convergence of the process with respect to $\|\cdot/q\|$-metrics and are used to prove an interval analogue of the Chibisov-O'Reilly theorem. A strong limit theorem related to the well-known Holder condition for Brownian bridge $U$ is also proved. Connections with related work of Csaki, Eicker, Jaeschke, and Stute are mentioned. As an application we introduce a new statistic for testing uniformity which is the natural interval analogue of the classical Anderson-Darling statistic.

Let $X_1, X_2, \cdots$, be a sequence of random variables and write $X^{(n)}_k$ for the $k$th largest among $X_1, X_2, \cdots, X_n$. If $\{k_n\}$ is a sequence of integers such that $k_n \rightarrow \infty, k_n/n \rightarrow 0$, the sequence $\{X^{(n)}_{k_n}\}$ is referred to as the sequence of intermediate order statistics corresponding to the intermediate rank sequence $\{k_n\}$. The possible limiting distributions for $X^{(n)}_{k_n}$ have been characterized (under mild restrictions) by various authors when the random variables $X_1, X_2, \cdots$ are independent and identically distributed. In this paper we consider the case when the $\{X_n\}$ form a stationary sequence and obtain a natural dependence restriction under which the "classical" limits still apply. It is shown in particular that the general dependence restriction applies to normal sequences when the covariance sequence $\{r_n\}$ converges to zero as fast as an appropriate power $n^{-\rho}$ as $n \rightarrow \infty$.

A Berry-Esseen bound is given for the rate of convergence to normality of the number of empty boxes when balls are distributed independently and at random to boxes with possibly unequal probabilities. The method of proof uses the equivalence of this distribution to a certain conditional distribution based on independent Poisson random variables. Then methods based on the characteristic function of this conditional distribution are used to obtain the result.

Convergence rates are derived in central limit theorems for martingale difference arrays. The rates depend heavily on the behavior of the conditional variances and on moment conditions. It is also shown that the rates which are obtained are the exact ones under the stated conditions.

We obtain the almost sure approximation of the partial sums of random variables with values in a separable Hilbert space and satisfying a strong mixing condition by a suitable Brownian motion. This is achieved by a modification of the proof of a similar result by Kuelbs and Philipp (1980) on $\phi$-mixing Banach space valued random variables. As by-products we get almost sure invariance principles for sums of absolutely regular sequences of random variables with values in a Banach space and necessary and sufficient conditions for the almost sure invariance principle for sums of independent, identically distributed random variables.

Let $\{X_n\}_{n \in \mathbb{Z}^d}$ be a stationary Gaussian process. It is proved that for all finite subsets $J$ of $\mathbb{Z}^d$ and complex-valued measurable functions $f_j, j \in J$, of a real variable, $|E(\prod_{j \in J}f_j(X_j))| \leq \prod_{j \in J}\|f_j(X_0)\|_p,$ where $p = \sum_{n \in \mathbb{Z}^d} \lbrack|E(X_0X_n)|/E(X^2_0)\rbrack$ is independent of $J$. A continuous version of this inequality is proved for stationary Gaussian processes $\{X_t\}_{t \in_\mathbb{R}^d}$. It is shown that for all bounded measurable subsets $\Lambda$ of $\mathbb{R}^d$ and complex-valued measurable functions $V$ of a real variable, $|E\big(\exp\big(\int_\Lambda V(X_t) d\mathbf{t}\big)\big)| \leq \|\exp(V(X_0))\|^{\|\Lambda\|}_p,$ where $|\Lambda|$ is the Lebesgue measure of $\Lambda$ and $p = \int_{\mathbb{R}^d} \lbrack|E(X_0X_t)|/E(X^2_0)\rbrack d\mathbf{t}$. Similar inequalities are proved for stationary Gaussian processes indexed by periodic quotient groups of $\mathbb{Z}^d$ and $\mathbb{R}^d$.

On the real line, there exist $\sigma$-finite measures which are not Radon measures, but are nevertheless defined on all bounded intervals $\big(\text{e.g.} \frac{1}{x} \sin \frac{1}{x} dx, \text{or} \sum_n\frac{(-1)^n}{n} \delta_{1/n}\big).$ Similarly, in stochastic calculus, there exist processes that, though not semimartingales, can be obtained as stochastic integrals of predictable processes with respect to semimartingales. This paper deals with such processes.

Let $B_t$ be a two-dimensional Brownian motion and $f(x)$ be a bounded measurable function vanishing outside a compact set. Then $(1/\lambda) \int^{e^{\lambda t}}_0 f(B_s) ds$ converges to const. $\ell(M^{-1}(t), 0)$ as $\lambda \rightarrow \infty$, where $\ell(t, x)$ and $M(t)$ are the local time and the maximum process of a one-dimensional Brownian motion, respectively. In the present article we generalize this theorem for more general Markov processes as follows: Let $X_t$ be a Markov process and $f(x)$ be a nonnegative, bounded measurable function on the state space. If the expectation of $\int^t_0 f(X_s) ds$ is asymptotically equal to a slowly varying function $L(t)$ as $t \rightarrow \infty$, then, $(1/\lambda) \int^{L -1(\lambda t)}_0 f(X_s) ds$ converges to $\ell(M^{-1}(t), 0)$ as $\lambda \rightarrow \infty$, in the sense of the convergence of all finite-dimensional marginal distributions.

The limiting distribution of the maximum term of the non-normal stationary sequence $\cdots X_{-1}, X_0, X_1 \cdots$ satisfying the autoregressive equation $X_n = \varepsilon_n + a_1X_{n-1} + a_2X_{n-2} + \cdots$ is investigated when $\sum |a_k| < 1$ and $\cdots \varepsilon_{-1}, \varepsilon_0, \varepsilon_1 \cdots$ are integrable real valued i.i.d. random variables having distributions with tails that are either Pareto or exponential in nature. Asymptotic results for the joint distribution of the first passage time $t = \inf\{n: X_n \geq c\}$ and the excess $R_t = X_t - c$ are also given as $c \rightarrow \infty$.

For $\cdots \varepsilon_{-1}, \varepsilon_0, \varepsilon_1 \cdots$ i.i.d. random variables the autoregression $X_n = \varepsilon_n + a_1X_{n-1} + a_2X_{n-2} + \cdots$ yields a payoff $\gamma^n \sum^n_{-\infty} w_kX_k$ when stopped at time $n, 0 < \gamma < 1$ being the discount factor. The optimal rule is characterized and under certain restrictions is the first passage time $t = \inf \{n: X_n \geq c\}$. As $c \rightarrow \infty$ the distributions of $t$ and the remainder term $R_t = X_t - c$ are asymptotically independent and determined for exponential and algebraic tailed distributions on $\varepsilon_n$. An asymptotic expression for the optimal payoff is given and $c = c(\gamma)$ is calculated so that $t$ yields a payoff asymptotically optimal and asymptotic to $c$ as $\gamma \rightarrow 1$.

Let $X_1, X_2, \cdots$, be a sequence of nonnegative i.i.d. random variables with common distribution $F$, and for each $n \geq 1$ let $X_{1n} \leq \cdots \leq X_{nn}$ denote the order statistics based on $X_1, \cdots, X_n$. Necessary and sufficient conditions are obtained for averages of the extreme values $X_{n+1-i, n}i = 1, \cdots, k_n + 1$ of the form: $k^{-1}_n \sum^{k_n}_{i = 1} (X_{n+1-i, n} - X_{n-k_n,n})$, where $k_n \rightarrow\infty$ and $n^{-1}k_n \rightarrow 0$, to converge in probability or almost surely to a finite positive constant. In the process, characterizations are given of the classes of distributions with regularly varying upper tails and of distributions with "exponential-like" upper tails.

The theory of positive dependence notions cannot yield useful results for some widely used distributions such as the multinomial, Dirichlet and the multivariate hypergeometric. Some conditions of negative dependence that are satisfied by these distributions and which have practical meaning are introduced. Useful inequalities for some widely used distributions are obtained.

Let $(T_1, T_2)$ and $(L_1, L_2)$ be two independent bivariate random vectors with distributions $F$ and $H$. Let $\tau_1 = \min(T_1, L_1), \tau_2 = \min(T_2, L_2)$ and let $G_{0,0}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, T_1 \leq L_1, T_2 \leq L_2\}$, $G_{0,1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, T_1 \leq L_1, L_2 < T_2\}, \leq L_2\}, G_{0, 1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, L_1 < T_1, T_2 \leq L_2\}$ and $G_{1,1}(s, t) = P\{\tau_1 \leq s, \tau_2 \leq t, L_1 < T_1, L_2 < T_2\}$. Under mild conditions the distributions $F$ and $H$ are expressed explicitly as functionals of $G_{0,0}, G_{0,0}, G_{1,0}$ and $G_{1,1}$. Necessary and sufficient conditions for the formulas to hold even when $(T_1, T_2)$ and $(L_1, L_2)$ are not independent are derived. Numerous applications are indicated. Extension of the results to $p$-dimensional distributions $(p > 2)$ is given.

Let $\{X_n\}$ be a sequence of independent identically distributed random variables which take the values $\pm 1$ with probability $\frac{1}{2}$. Let $X = \sum^\infty_{n=1} a_nX_n$ where $\sum a^2_n < \infty$. We show that if $n^{_\alpha} \leq |a_n| \leq n^{-\beta}$ for some $\alpha > \frac{1}{2}$ and $0 \leq \alpha - \beta < \frac{1}{2}$ then the distribution of $X = \sum a_nX_n$ is absolutely continuous with respect to Lebesgue measure. We then prove similar results for more general independent sequences. We also show that if $\lim\inf 2^N \sqrt{\sum^\infty_n=N+1} a^2_n = 0$ then the distribution of $X = \sum a_nX_n$ is singular with respect to Lebesgue measure.

Let $\{X_n\}$ be a sequence of independent random variables and $\{a_n\}$ a positive decreasing sequence such that $\sum a_nX_n$ is a random variable. We show that under mild conditions on $\{X_n\}$ (i) if for every $\delta, \lambda > 0$ $\sum^\infty_{n=1} \int^{\delta/a_{n+1}}_{\delta/a_n} \exp(-\lambda\xi^2 \sum^\infty_{k=n+1} a^2_k) d\xi < \infty$ then $P(\sum a_nX_n \in dx)$ has a density. (ii) $\lim_{\xi \rightarrow \infty}|E(e^{i\xi\sum a_nX_n})| = 0$ for every $\{X_n\} \operatorname{iff} \lim_{N\rightarrow\infty} a^{-2}_N \sum^\infty_{n=N + 1} a^2_n = \infty.$ Several consequences, examples and counterexamples are given.

Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.

Sufficient conditions in terms of interpolation variances are given for a Gaussian process to have a jointly continuous local time. In the stationary case these conditions can be verified in terms of the spectral density and are seen to be within logarithmic factors of the best possible conditions. A bound for the modulus of continuity in the space variable is also obtained.

Continuity in the time parameter is considered for a natural version of the local time for stationary Gaussian processes. Bounds are given for the local and uniform modulus of continuity which are applicable in cases not covered by Kono, notably when the incremental variance is a regularly varying function of index two.

Consider random variables $x_1, \cdots, x_n$, independently and uniformly distributed on the unit interval. Suppose we are given partial information, $\Gamma$, about the unknown ordering of the $x$'s; e.g., $\Gamma = \{x_1 < x_{12}, x_7 < x_5, \cdots\}$. We prove the "XYZ conjecture" (originally due to Ivan Rival, Bill Sands, and extended by Peter Winkler, R. L. Graham, and other participants of the Symposium on Ordered Sets at Banff, 1981) that $P(x_1 < x_2|\Gamma) \leq P(x_1 < x_2|\Gamma, x_1 < x_3).$ The proof is based on the FKG inequality for correlations and shows by example that even when the hypothesis of the FKG inequality fails it may be possible to redefine the partial ordering so that the conclusion of the FKG inequality still holds.

Suppose that for every independent sequence of random variables satisfying some hypothesis condition $H$, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition $H$. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences.

In a recent paper Barlow and Proschan noted that similar independence results appeared both in life table analysis and in fixed interval analysis. In this note we present a general asymptotic independence result of which these results are special cases. Also some further applications are given. In essence, our method may be described as follows: Assume that we are interested in $g$ quantities $q_1, \cdots, q_g$, each constructed from a sequence of independent random variables, and that these sequences are conditionally independent given their random lengths. Then by the completion of each sequence with independent random variables we obtain $g$ independent sequences. Under rather general assumptions we are able to deduce asymptotic properties of the original sequence from the corresponding properties of the completed sequences. In particular, we are often able to prove the asymptotic independence of the quantities $q_1, \cdots, q_g$.

For an urn model (arising typically in the sequential estimation of the size of a finite population), along with an invariance principle for a partial sequence of nonnegative random variables, a renewal theorem relating to some stopping times is established. A representation of these random variables in terms of linear combinations of some martingale-differences provides the key to a simple solution.

An asymptotic expansion for the expected sample size in the non-linear renewal theorem is shown to hold under an alternative set of regularity conditions. The alternative set requires fewer moments than existing conditions in an important special case, but is more restrictive in other ways.

In 1972, K. Urbanik introduced the notion of operator-selfdecomposable probability measures (originally they were called Levy's measures). These measures are identified as limit distributions of partial sums of independent Banach space-valued random vectors normed by linear bounded operators. Recently, S. J. Wolfe has characterized the operator-selfdecomposable measures among the infinitely divisible ones. In this note we find examples of measures whose finite convolutions are a dense subset in a class of all operator-selfdecomposable ones.

The author expresses the distributions of stable laws of index $2^{-n}$ in terms of the normal and Chi squared distributions. This paper is an extension of an earlier result obtained by the author when he considered symmetric cases only.

Let $A$ be a $C^2$ curve of length $L(A)$ in some Euclidean space. Let $P_n$ be a sequence of randomly chosen polygons with $n$ vertices which are inscribed in $A$. It is shown that with probability 1 $\lim n^2\lbrack L(A) - L(P_n)\rbrack = \frac{1}{4} \int_A \kappa^2(s) ds$ where $\kappa$ is the curvature.

Let $X_1, X_2, \cdots$ be a sequence of independent uniformly distributed random variables on $\lbrack 0, 1\rbrack$ and $K_n$ be the $k$th largest spacing induced by $X_1, \cdots, X_n$. We show that $P(K_n \leq (\log n - \log_3n - \log 2)/n$ i.o.) = 1 where $\log_j$ is the $j$ times iterated logarithm. This settles a question left open in Devroye (1981). Thus, we have $\lim \inf(nK_n - \log n + \log_3n) = -\log 2 \text{almost surely},$ and $\lim \sup(nK_n - \log n)/2 \log_2n = 1/k \text{almost surely}$.

The rates at which a transient random walk can escape to infinity are discussed.

A prior distribution $\wedge$ on a set of parameters $I$ is said to be ergodically decomposable if $\wedge$ - a. all probability measures $(\pi_i)_{i\in I}$ are mutually singular in some strong sense. Criteria are established for $\wedge$ to be ergodically decomposable in terms of the posterior distribution and the Bayes estimator, which, for $I$ the locally finite measures on a Polish space and $\pi_i$ the Poisson process with intensity $i$, is just the Papangelou kernel of the Cox process directed by $\wedge$.

We solve a problem posed by J. P. Burgess and R. D. Mauldin, and thus show that a result of theirs on conditional distributions and orthogonal measures cannot be improved.

A family $\{\mu_\alpha\}$ of measures on a $\sigma$-field $\mathscr{B}$ on a space $X$ is "uniformly orthogonal" means that for each $\alpha, \exists H_\alpha \in \mathscr{B}$ such that $\mu_\alpha (X - H_\alpha) = \mu_\beta(H_\alpha) = 0$ if $\beta \neq \alpha$. Assuming $CH$, an example is given of an orthogonal family of measures on the Borel sets of $I^2$ such that no uncountable subfamily is uniformly orthogonal. Assuming $\sim CH + MA$, such an uncountable family obviously exists.