The paper is devoted to a study of the exit boundary of random walks on discrete groups and related topics. We give an entropic criterion for triviality of the boundary and prove an analogue of Shannon's theorem for entropy, obtain a boundary triviality criterion in terms of the limit behavior of convolutions and prove a conjecture of Furstenberg about existence of a nondegenerate measure with trivial boundary on any amenable group. We directly connect Kesten's and Folner's amenability criteria by consideration of the spectral measure of the Markov transition operator. Finally we give various examples, some of which disprove some old conjectures.

Let $X_1, X_2, \cdots$ be i.i.d. mean zero random variables. Let $S_n = X_1 + \cdots + X_n$ and $M_\varepsilon = \sup_{n\geqq 1} (S_n - n\varepsilon)^+$ for $\varepsilon > 0$. Suppose $\sigma^2 = E(X_1)^2$ is positive and finite. Then $EM_\varepsilon < \infty$ and $2\varepsilon\sigma^{-2} EM_\varepsilon$ converges to 1 as $\varepsilon \searrow 0^+$. In this paper we obtain an approximation of the discrepancy $1 - 2\varepsilon\sigma^{-2} EM_\varepsilon$ as $\varepsilon \searrow 0^+$. To do so we derive a first order approximation of $P(M_\varepsilon < y)$ which is uniform in $y$ as $\varepsilon \searrow 0^+$ and asymptotically exact for $y$ on $\lbrack y_\varepsilon, \infty)$ provided $y_\varepsilon \rightarrow \infty$. Approximation of $P(M_\varepsilon < y)$ necessitates a digression into renewal theory. We derive an approximation of the expected time $E\tau_y$ required by a sum $T_n = Y_1 + \cdots + Y_n$ of i.i.d. non-negative random variables to reach or exceed $y$. The bounds obtained are of particular interest when $EX = \infty$ and are best possible in a rather strong sense.

We examine the relationship between the central limit theorem with Gaussian limit, and law of the iterated logarithm type behavior in the Banach space setting. Our results solve some problems posed by Kesten (1972) and generalize some results of Klass (1976, 1977).

Let $S_n, n = 1,2,\cdots$, denote the partial sums of a stationary $m$-dependent sequence of random variables with positive expectation. The first passage times $\min\{n: S_n > t\}$ are investigated. Several results are extended from the case of independent variables.

For simplicity, let $d = 2$ and consider the points $(n, m)$ in $Z^2_+$, with $\theta m \leq n \leq \theta^{-1}m$, where $0 < \theta < 1$. For i.i.d. random variables with this set as an index set we present a law of the iterated logarithm, strong laws of large numbers and related results. We also observe that (and try to explain why) the martingale proof of the Kolmogorov strong law of large numbers yields a weaker result for this index set than the classical proofs, whereas this is not the case if the index set is all of $Z^d_+, d \geq 1$.

Let $X_t$ be a symmetric stable process of index $\alpha$ in $\mathbb{R}^n$ and $\tau = \inf\{t: X_t \not\in D\}$ where $D$ is a connected open region in $\mathbb{R}^n$. If $0 < p < \alpha$ two sided $L^p$ inequalities are obtained between $\tau^{1/\alpha}$ and the maximal function $X^\ast_\tau = \sup_{t < \tau} |X_t|$. Analytic conditions for $\tau^{1/\alpha} \in L^p$ are given in terms of domination of $|x|^p, x \in D^c$ by a function $u(x) \alpha$-harmonic in $D$. Also, the boundary behavior of $\alpha$-harmonic functions is studied by obtaining two-sided $L^p$ inequalities, $0 < p < \infty$, between a random and deterministic maximal function of non-negative $\alpha$-harmonic functions.

For transient symmetric Levy processes we determine a uniform lower bound for the Hausdorff dimension of the range of a process on various time sets. This complements earlier work which provided a uniform upper bound. An example is provided in which both bounds are attained.

Let $W(t)$ be a standard Wiener process with local time (occupation density) $L(x, t)$. Kesten showed that $L(0, t)$ and $\sup_x L(x, t)$ have the same LIL law as $W(t)$. Exploiting a famous theorem of P. Levy, namely that $\{\sup_{0 \leq s \leq t} W(s), t \geq 0\} {\underline{\underline{\mathscr{D}}}} \{L(0, t), t \geq 0\}$, we study the almost sure behaviour of big increments of $L(0, t)$ in $t$. The very same increment problems in $t$ of $L(x, t)$ are also studied uniformly in $x$. The results in the latter case are slightly different from those concerning $L(0, t)$, and they coincide only for Kesten's above mentioned LIL.

Let $W(t)$ be a standardized Wiener process. In this paper we prove that $\lim \sup_{T\rightarrow\infty} \max_{a_T \leq t \leq T}\frac{|W(T) - W(T - t)|}{\{2t\lbrack\log(T/t) + \log \log t\rbrack\}^{1/2}} = 1 \text{a.s.}$ under suitable conditions on $a_T$. In addition we prove various other related results all of which are related to earlier work by Csorgo and Revesz. Let $\{X_k\}$ be an i.i.d. sequence of random variables and let $S_N = X_1 + \cdots + X_N$. Our original objective was to obtain results similar to the ones obtained for the Wiener process but with $N$ replacing $T$ and $S_N$ replacing $W(T)$. Using the work of Komlos, Major, and Tusnady on the invariance principle, we obtain the desired results for i.i.d. sequences as immediate corollaries to our work for the Wiener process.

Harmonic functions and the Dirichlet problem on an open set $G$ are defined for a pieced-out or revival Markov process constructed from a continuous base process. A one-to-one correspondence is obtained between the bounded harmonic functions of the revival process and those of the base process. The harmonic functions of the revival process are shown to be continuous on $G$ under certain conditions and to coincide with the solutions of $\mathscr{U}f = 0$ on $G$ where $\mathscr{U}$ is the characteristic operator. These results are applied to random evolution processes and branching processes.

By extending ideas and methods of Dudley (1977) (who was dealing with the Brownian case), we prove that a necessary and sufficient condition for all martingales of a given filtration $(\mathscr{F}_t)$ to be continuous, is that, for every stopping time $T$ and every $\mathscr{F}_T$-measurable random variable $X$, there exists a continuous local martingale $M$ with $M_T = X$ a.s. Moreover, $M$ can be chosen such that $M_0 = 0$ on a reasonably large event (equal to $\{T > 0\}$ in the Brownian case); if there exists a Brownian motion $B$ adapted to $(\mathscr{F}_t), M$ can be chosen as a stochastic integral of some $(\mathscr{F}_t)$-predictable process with respect to $B$ (even when $(\mathscr{F}_t)$ is larger than the natural filtration of $B$).

We define the notion of a Markov strategy for the general optimal control problem where the index set is partially ordered. We prove that the supremum over all strategies is always attained by a Markov strategy if and only if the structure of the probability space is that of a Markov random field.

Let $\mu$ be an invariant measure for a Markov process which is assumed $\mu$-uniformly ergodic in the following sense: the corresponding semigroup of operators on $L^2(d\mu)$, say $\{P_t; t \geq 0\}$, is such that the time average $(1/T) \int^T_0 P_t dt$ converges to a rank one projection in the uniform norm of operators. We prove that hitting times of sets having non zero $\mu$-measure possess moment generating functions.

Let $(\Omega, \mathscr{J}, P; \mathscr{J}_{s,t})$ be a probability space with a family of sub-$\sigma$-algebras indexed by $(s, t) \in \lbrack 0, \infty) \times \lbrack 0, \infty)$, satisfying the usual conditions. Let $X(s,t)$ be a solution of a stochastic differential equation in the plane with respect to the Wiener-Yeh process. Under one of the usual conditions used to guarantee existence and uniqueness of a solution to the equation, it is shown that the absolute moments of $X(s,t)$ grow at most exponentially in $st$. The estimate is based on a version of the two parameter Ito formula and on an extension of Gronwall's inequality to functions of two variables.

The two following results are proved. Given $(\Omega, \mathscr{F}, \mu, T)$ where $T$ is a measurable transformation preserving the probability measure $\mu$, given a sequence $f_n$ of integrable functions such that $\int (f_{n+k} - f_n - T^nf_k)^+ d\mu \leq c_k \text{with} \lim_k \frac{1}{k} c_k = 0,$ then $(1/n) f_n$ is converging in $L^1$-norm. If, furthermore, $f_{n+k} - f_n - T^nf_k \leq T^nh_k$ with $h_k$ a sequence of positive functions whose integrals are bounded, then $(1/n) f_n$ is also converging a.e. From this extension of Kingman's subadditive ergodic theorem, the Shannon-McMillan-Breiman theorem follows at once.

Stationarity is a property of infinite sequences of random variables. An appropriate extension of this definition is made, to cover finite sequences. The set of finite stationary sequences is shown to be a convex set and its extreme points are related to shift register sequences (which are paths on a graph known as the shift net, or the de Bruijn graph). The set of finite stationary sequences as defined here is simply the set of finite dimensional projections of infinite stationary sequences.

Let $(\mu_t)_{t>0}$ be a Gaussian semigroup on a metric, separable, complete group $G$. If $H$ is a Borel measurable normal subgroup of $G$ such that $\mu_t(H) > 0$ for all $t$, then $\mu_t(H) = 1$ for every $t$. If, in addition, $\mu_t$ are symmetric, then $\mu_t(H) > 0$ for a single $t$ implies $\mu_t(H) = 1$ for all $t$.

We study some $\{0, 1, \cdots\}^{z^d}$ valued Markov interactions $\eta_t$ called contact path processes. These are similar to branching random walks, in that the normalized size process starting from a singleton is a martingale which converges to a limit $M_\infty$. In contrast to branching, however, $M_\infty$ depends on the spatial dynamics of the path process. The main result is an exact evaluation of the variance of $M_\infty$, achieved by means of the Feynman-Kac formula. The basic contact process of Harris may be viewed as a projection of $\eta_t$; as a corollary to the main result we obtain bounds on the contact process critical value $\lambda^{(d)}_c$ in dimension $d \geq 3$.

Consider a system of identical particles moving on the integer lattice with mutual annihilation of any pair of particles which collide. Apart from this interference, all particles move independently according to the same random walk $p$. A system will be called site recurrent if a.s. each site is occupied at arbitrarily large times. The following generalization of a conjecture by Erdos and Ney was open: the system of annihilating simple random walks on $Z_2$, starting with all sites except the origin occupied, is site recurrent. We prove, for general $p$ and a reasonably broad class of initial distributions, that the annihilating system is site recurrent. Loosely speaking, this condition is that the initial configuration does not have any fixed sequence of holes with diameters tending to infinity.

Finite nearest particle systems are certain one parameter families of continuous time Markov chains $A_t$ whose state space is the collection of all finite subsets of the integers. Points are added to or taken away from $A_t$ at rates which have a particular form. The empty set is absorbing for these chains. In the reversible case, the parameter $\lambda$ is normalized so that extinction at the empty set is certain if and only if $\lambda \leq 1$. Let $\sigma(\lambda)$ be the probability of nonextinction starting from a singleton. In a recent paper, Griffeath and Liggett obtained the bounds $\lambda^{-1}(\lambda - 1) \leq \sigma(\lambda) \leq |\log \lambda^{-1}(\lambda -1)|^{-1}$ for $\lambda > 1$, and raised the question of determining the correct asymptotics of $\sigma(\lambda)$ as $\lambda \downarrow 1$. In the present paper, this question is largely answered by showing under a moment assumption that for $\lambda > 1, \sigma(\lambda)$ is bounded above by a constant multiple of $\lambda - 1$. In the critical case $\lambda = 1$, a similar improvement is made on the known bounds on the asymptotics as $n \rightarrow \infty$ of the probability that $A_t$ is of cardinality at least $n$ sometime before extinction. Similar results have been conjectured, but remain open problems in nonreversible situations--for example, for the basic one-dimensional contact process.

We give new results on the total variation distance of the distribution of a point process on the line from that of a Poisson process. Both one dimensional and function space distances are considered. Additionally similar bounds for marked point processes are given, both for the finite and infinite mark space cases. The bounds are all given in terms of the compensator of the point process (with respect to an arbitrary filtration) and are analogues and extensions of discrete time results of Freedman (1974) and Serfling (1975). Some new techniques for discrete approximation of compensators are used in the proofs. Examples of the use of the bounds appear elsewhere (Brown and Pollett, 1982, Brown, 1981), but an application to compound Poisson approximation and thinning of point processes is given here.

A class of empirical processes having the structure of $U$-statistics is considered. The weak convergence of the processes to a continuous Gaussian process is proved in weighted sup-norm metrics stronger than the uniform topology. As an application, a central limit theorem is derived for a very general class of non-parametric statistics.

Homogeneous nondecreasing functions of independent NBU random variables are studied. Two results of Block and Savits are improved. It is shown that if a coherent system, formed from independent NBU components, has exponential life then it is essentially a series system with exponential components. Also, it is shown that if a strictly increasing homogeneous function of independent NBU random variables has an exponential distribution then it is essentially a univariate function of one of its variables which must, then, be exponential. A new characterization of the MNBU class of distributions of Marshall and Shaked is derived, and a new proof of the closure of the class of NBU distributions under formation of nonnegative homogeneous increasing functions is given.

A system with $n$ independent components which functions if and only if at least $k$ of the components function is a $k$ out of $n$ system. Parallel systems are 1 out of $n$ systems and series systems are $n$ out of $n$ systems. If $\mathbf{p} = (p_1, \cdots, p_n)$ is the vector of component reliabilities for the $n$ components, then $h_k(\mathbf{p})$ is the reliability function of the system. It is shown that $h_k(\mathbf{p})$ is Schur-convex in $\lbrack(k - 1)/(n - 1), 1\rbrack^n$ and Schur-concave in $\lbrack 0, (k - 1)/(n - 1)\rbrack^n$. More particularly if $\prod$ is an $n \times n$ doubly stochastic matrix, then $h_k(\mathbf{p}) \geq (\leq)h_k(\mathbf{p}\prod)$ whenever $\mathbf{p} \in \lbrack(k - 1)/(n - 1), 1\rbrack^n(\lbrack 0, (k - 1)/(n - 1)\rbrack^n)$. This Theorem is compared with a result on Schur-convexity and -concavity by Gleser [2] which in turn extends work of Hoeffding [4].

We determine the exact optimal bounds $A_p$ and $B_p$ such that $A_p\lbrack E|X|^p + E|Y|^p\rbrack \leq E|X + Y|^p \leq B_p\lbrack E|X|^p + E|Y|^p\rbrack,$ $(p \geq 1)$, whenever $X, Y$ are i.i.d. random variables with mean zero. We give examples of random variables attaining equality or nearly so. Exactly the same bounds $A_p$ and $B_p$ are obtained in the more general case where it is only assumed that $E(X \mid Y) = E(Y \mid X) = 0$.

For a class of expansive transformations of the unit interval, we show a local limit theorem for the process $(f \circ T^n)_{n \in N}$, where $f$ is a real bounded variation function. We show also, that the speed in the central limit theorem is $1/\sqrt n$.

Let $\{S_i\}, i = 1, \cdots, n$, be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs. Let $f$ be a rotation of the circle and let $u$ denote Lebesgue measure. Then the matrix $P$ defined by $p_{ij} = u(S_i \cap f^{-1} S_j)/u(S_i)$ is stochastic. We prove (and improve) a conjecture of Joel E. Cohen asserting that every irreducible stochastic matrix arises from a construction of this type.

Let $\{X_n\}$ be random elements in a separable Banach space which is $p$-smoothable and let $\{a_k\}$ and $\{A_k\}$ denote positive random variables such that almost surely $A_k$ is monotonically increasing to $\infty$ and that $A_k/a_k \rightarrow \infty$. Convergence almost surely is obtained for the weighted sum $A^{-1}_n \sum^n_{k=1} a_kX_k$ and is related to a moment condition on the random elements and a growth condition on the random weights.

We give a concise proof, using rescaling, random time change and simple infinitesimal generator computations of the fact that Brownian excursion is the weak limit as $\varepsilon \rightarrow 0$ of tied down Brownian motion conditioned never to fall below $-\varepsilon$.

We show that the finiteness of the moment generating function is necessary for the finiteness of the lim sup of the moving averages considered by Shepp (1964). This also implies that the same must be true for the Erdos-Renyi law of large numbers.

Let $W(t), 0 \leq t \leq 1$, be the Wiener process tied down at $t = 0, t = 1; W(0) = W(1) = 0$. We find the distribution of $\sup_{0 \leq t \leq 1} W(t) - \int^1_0 W(t) dt$ in terms of the zeros of the Airy function and the positive stable density of exponent 2/3. This corresponds to the distribution of the supremum of a certain stationary, mean zero, periodic Gaussian process. It is also the limiting distribution of an optimal test statistic for the isotropy of a set of directions, proposed by G. S. Watson.

Let $X(t), 0 \leq t \leq 1$, be the Brownian bridge and $L = \int^1_0 |X(t)| dt$. Using results of Shepp [5] and Rice [4], an explicit formula for $P(L \leq \ell)$ is obtained.

For every sequence $(\varepsilon_n)_{n \in N}$ in (0, 1) there exists a strictly stationary orthonormal sequence $(X_n)_{n \in N}$ of random variables with $|P(A \cap B) - P(A)P(B)| \leq \varepsilon_n$ for all $A \in \sigma(X_1, \cdots, X_k), B \in \sigma(X_{k+n}, X_{k+n+1}, \cdots), k \in \mathbb{N}, n \in \mathbb{N}$, such that the distribution of $n^{-1/2} \sum^n_{i=1} X_i$ is not weakly convergent to the standard normal distribution.

If the sequence of coefficients of a random power series has a degenerate tail field, then either its circle of convergence is a natural boundary, or this situation can be achieved by subtracting a fixed series. This generalises the known result for independent coefficient sequences.

In this paper we derive a new explicit formula for the Laplace transform of the length of the busy period for the $M|G|1$ queue by a direct martingale method of independent interest. The method is probabilistic, of general character and avoids tedious calculations with complex variables.

Given probabilities $\mu$ and $\nu$ on $(X, \mathscr{A})$ and $(X, \mathscr{B})$ respectively, a probability $\eta$ on $(X, \mathscr{A} \vee \mathscr{B})$ is called a splicing of $\mu$ and $\nu$ if $\eta(A \cap B) = \mu(A) \nu(B)$ for all $A \in \mathscr{A}, B \in \mathscr{B}$. Using a result of Marczewski we give an elementary proof of Stroock's result on the existence of splicing. We also discuss the splicing problem when $\mu$ and $\nu$ are compact measures.

It is well known that the smaller and the larger of a random sample of size two are positively correlated. The coefficient of correlation is at most one-half, and the upper bound is attained only for rectangular distributions.

Let $\{\xi_k: k \geq 0\}$ be an orthogonal sequence of random variables with finite second moments $E\xi^2_k = \sigma^2_k$. It is well-known that if $\sum^\infty_{k=0} \sigma^2_k(k + 1)^{-2}\lbrack\log(k + 2)\rbrack^2 < \infty$, then the first arithmetic means $\tau^0_n: = (n + 1)^{-1} \sum^n_{k=0} \xi_k \rightarrow 0$ a.s. $(n \rightarrow \infty)$. Now we prove that the means $\tau^1_n: = (n + 1)^{-1} \sum^n_{k=0} (1 - k(n + 1)^{-1})\xi_k \rightarrow 0$ a.s. $(n \rightarrow \infty)$ merely under the condition $\sum^\infty_{k=0} \sigma^2_k(k + 1)^{-2} < \infty$. We define the means $\tau^\alpha_n$ for every real $\alpha$, too and prove that under the latter condition $\tau^\alpha_n \rightarrow 0$ a.s. $(n \rightarrow \infty)$ provided $\alpha > 0$.