The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but some departures from stationarity are considered. Significant ideas and methods are described rather than details, and, in particular, the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of processes (e.g., normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

In this paper we prove a rate of convergence result for the density of normalized sample maxima to the appropriate limit density. Also a local limit theorem--with rates of convergence--for maxima of i.i.d. random variables is proved.

For random walks $\{S_n\}$ whose distribution can be embedded in an exponential family, large-deviation approximations are obtained for the probability that $\max_{0\leq i < j\leq m}(S_j - S_i) \geq b$ (i) conditionally given $S_m$ and (ii) unconditionally. The method used in the conditional case seems applicable to maxima of a reasonably large class of random fields. For the unconditional probability a more special argument is used, and more precise results obtained.

If $\{Z_n\}^\infty_0$ is a critical branching process such that $E_1Z^2_1 < \infty$, then $(\log n)^{-1}E_iM_n \rightarrow i$, where $E_i$ refers to starting with $Z_0 = i$ and $M_n = \max_{0\leq j \leq n}Z_j$. This improves the earlier results of Weiner [9] and Pakes [7].

Ryll-Nardzewski has proved that an infinite sequence of random variables is exchangeable if every subsequence has the same distribution. We discuss some restatements and extensions of this result in terms of martingales and stopping times. In the other direction, we show that the distribution of a finite or infinite exchangeable sequence is invariant under sampling by means of a.s. distinct (but not necessarily ordered) predictable stopping times. Both types of result generalize to exchangeable processes in continuous time.

Consider random letter sequences $\{\xi^{(\sigma)}_t, t = 1,\ldots, N; \sigma = 1,\ldots, s\}$ based on a finite alphabet generated by uniformly mixing stationary processes. The asymptotic distributional properties of the length of the longest common word in $r$ or more of the $s$ sequences $K_{r,s}(N)$, are investigated. When the probability measures of the different sequences are not too dissimilar, a classical extremal type limit law holds for $K_{r,s}(N) - (r \log N/(-\log \lambda)), \lambda$ being an appropriate local match parameter. The distributional properties of other long-word relationships and patterns among the sequences are also discussed.

We construct the Kuznetsov measure associated with an excessive measure, $m$, and the semigroup of a process killed according to a multiplicative functional, $M$, in terms of the Kuznetsov measure corresponding to $m$ and the original semigroup and certain functionals coming from $M$. We also obtain an explicit decomposition of $m$ into its invariant and purely excessive parts relative to the killed semigroup. This decomposition involves objects related to the fine structure of $M$ that have been introduced in recent years.

We study a random time change for stationary Markov processes $(Y_t, Q)$ with random birth and death. We use an increasing process, obtained from a homogeneous random measure (HRM) as our clock. We construct a time change that preserves both the stationarity and the Markov property. The one-dimensional distribution of the time-changed process is the characteristic measure $\nu$ of the HRM, and its semigroup $(\tilde{P}_t)$ is a naturally defined time-changed semigroup. Properties of $\nu$ as an excessive measure for $(\tilde{P}_t)$ are deduced from the behaviour of the HRM near the birth time. In the last section we apply our results to a simple HRM and connect the study of $Y$ near the birth time to the classical Martin entrance boundary theory.

A two-parameter Markov process $X$ with regular trajectories is associated to a pair of commuting Feller semigroups $P^1$ and $P^2$ considered on the same space $E$. A subsequent potential theory is developed with respect to an operator $\mathscr{L}$ which is the product of the generators of $P^1$ and $P^2$, respectively. The definition of a harmonic function $f$ on an open subset $A$ is expressed in terms of the hitting stopping line of $A^c$ by $X$ and the stochastic measure generated by $f(X)$. A PDE problem in $A$ with boundary conditions on $A^c$ is studied.

Time reversal of semimartingales defined on a Levy process framework is considered. Usually semimartingales cannot be time-reversed such that the reversed process is still a semimartingale. An expansion-of-filtrations result for Levy processes is established and then it is used to give sufficient conditions such that a semimartingale defined on a Levy process can be time-reversed and still remain a semimartingale.

We describe a general class of multivariate infinitely divisible distributions and their related stochastic processes. Then we prove inequalities which are the analogs of Slepian's inequality for these distributions. These inequalities are applied to the distributions of $M/G/\infty$ queues and of sample cumulative distribution functions for independent multivariate random variables.

This paper proves a converse to a theorem of L. D. Brown and Y. Rinott concerning positive dependence ordering for multivariate compound Poisson distributions.

Let $L = \frac{1}{2} \nabla \cdot a\nabla + b \cdot \nabla$ generate a diffusion process on $R^d$. An expression involving $a$ and $b$ on $1 \leq |x| \leq n$ and two functions $g$ and $h$, varied over suitable domains, attains its mini-max value at $\lambda_n$. It is shown that $\lim_{n\rightarrow\infty}\lambda_n = 0$ or $\lim_{n\rightarrow\infty} \lambda_n > 0$ according to whether the process is recurrent or transient.

Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent and identically distributed random variables and let $m_n$ and $k_n$ be positive integers such that $m_n \rightarrow \infty, k_n \rightarrow \infty, m_n/n \rightarrow 0$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. We find a necessary and sufficient condition for the existence of normalizing and centering constants $A_n > 0$ and $B_n$ such that the sequence $T_n = A^{-1}_n \big\{\sum_{i=m_n+1}^{n-k_n} X_{i,n} - B_n\big\}$ is stochastically compact and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $B_n$ such that $T_n$ is asymptotically normal. A variant of Stigler's theorem when $m_n/n \rightarrow \alpha$ and $k_n/n \rightarrow 1 - \beta$, where $0 < \alpha < \beta < 1$, is also obtained as a by-product.

A central limit theorem for interacting diffusion processes is shown. The proof is based on an infinite-dimensional stochastic integral representation of smooth functionals of diffusion processes. Exponential decay of correlations and the equation of the fluctuation field are also obtained.

We prove a correlation inequality concerning the occupation of disjoint sets in the symmetric exclusion process. As an application we derive a pointwise ergodic theorem for this same process.

We prove a zero or one law for one dimensional random walks in random environments for which the probability of making jumps of size $n$ decays exponentially. As an application we conclude that these random walks are recurrent if the distribution of the random environment is symmetric.

We investigate the smoothness properties of the solutions of one-dimensional wave equations with nonlinear random forcing. We define singularities as anomalies in the local modulus of continuity of the solutions. We prove the existence of such singularities and their propagation along the characteristic curves. When the space variable is restricted to a bounded interval, we impose the Dirichlet boundary condition at the endpoints and we show how the singularities are reflected at the boundary.

Suppose $\Phi = \{\phi_a; a \in A\}$ is a Gaussian random field. Let $\rho$ be a function on the parameter set $A$ with values in an open interval $I$. To every $t$ in $I$, there corresponds a subfield $\Phi_t = \{\phi_a; \rho(a) = t\}$ of the field $\Phi$. The family $\Phi_t, t \in I$, can be viewed as a Gaussian stochastic process. With a proper modification, this setup can be applied to generalized random fields for which the values at single points are not defined, in particular to the free field. In the case of a linear function $\rho$, the Gaussian process $\Phi_t$ plays a fundamental role in quantum field theory. It is a stationary Gaussian Markov process, where its Markov semigroup is given by the Feynman-Kac-Nelson formula. We prove that for a wide class of functions $\rho, \Phi_t$ is a nonhomogeneous Markov process and we evaluate the generators of this process.

A construction of multiple stochastic integrals with respect to a strictly $p$-stable random measure is given, $0 < p \leq 2$. The integrands are Banach space-valued deterministic functions.

In this paper we give the local time for some continuous two-parameter martingales with respect to the quadratic variation $\langle M\rangle$ and we study some of their sample path properties.

Let $T$ be the first time that a perturbed random walk crosses a nonlinear boundary. This paper concerns the approximations of the distribution of the excess over the boundary, the expected stopping time $ET$ and the variance of the stopping time $\operatorname{Var}(T)$. Expansions are obtained by using linear renewal theorems with varying drift.

For normal random walks $S_1, S_2,\ldots$, formed from independent identically distributed random variables $X_1, X_2,\ldots$, we determine the asymptotic behavior under regularity conditions of $P(S_n > mg(n/m) \text{for some} n < m\mid S_m = m\xi_0, U_m = m\lambda_0), \quad\xi_0 < g(1),$ where $U_m = X^2_1 + \cdots + X^2_m$. The result is applied to a normal change-point problem to approximate null distributions of test statistics and to obtain approximate confidence sets for the change-point.

Let $X_1, X_2,\ldots$ be independent random elements taking values in a Banach space $(B, \|\cdot\|)$ and having partial sums $S_n = X_1 + \cdots + X_n$. Let $\alpha > 0$ and let $\Phi: \lbrack 0, \infty)$ be a nondecreasing continuous function such that $\Phi(0) = 0$ and $\Phi(cx) \leq c^\alpha\Phi(x)$ for all $c \geq 2, x \geq 0$. Put $a^\ast_n = E \max_{1\leq k\leq n}\Phi(\|S_k\|)$. Let $T$ be any (possibly randomized) stopping time w.r.t. $\{S_n\}$. We prove that $E \max_{1\leq n\leq T}\Phi(\|S_n\|) \leq 20(18^\alpha)Ea^\ast_T$. If $\{S_n\}$ is a mean-zero $B$-valued martingale and $\lim_{n\rightarrow\infty}E\|S_{T\wedge n}\| < \infty$, it is shown that $L \equiv \lim_{n\rightarrow\infty}ES_nI(T > n)$ always exists and $ES_T = -L$, so that $ES_T = 0 \operatorname{iff} L = 0$. Let $s_n = E\|S_n\|$ and $s^\ast_n = E \max_{1\leq k\leq n}\|S_k\|$. As a consequence of these facts it follows that if $\{X_n\}$ are independent and have mean zero, then $E\|S_T\| < \infty$ and $ES_T = 0$ whenever $Es^\ast_T < \infty$. In the mean-zero case $s^\ast_n \leq 4s_n$; and so, in fact, $Es_T < \infty$ implies $ES_T = 0$. This constitutes a best possible improvement of Wald's equation.

A sharp bound for the dependence between sets of order statistics is established which gives precise information about the asymptotic independence of order statistics.

This article proves that the spectral distribution of the random matrix $(1/2\sqrt{np}) (X_pX'_p)$, where $X_p = \lbrack X_{ij}\rbrack_{p\times n}$ and $\lbrack X_{ij}: i, j = 1,2,\ldots\rbrack$ has iid entries with $EX^4_{11} < \infty, \operatorname{Var}(X_{11}) = 1$, tends to the semicircle law as $p \rightarrow \infty, p/n \rightarrow 0$, a.s.

We ask how an investor (with knowledge of the past) should distribute his funds over various investment opportunities to maximize the growth rate of his compounded capital. Breiman (1961) answered this question when the stock returns for successive periods are independent, identically distributed random vectors. We prove that maximizing conditionally expected log return given currently available information at each stage is asymptotically optimum, with no restrictions on the distribution of the market process. If the market is stationary ergodic, then the maximum capital growth rate is shown to be a constant almost surely equal to the maximum expected log return given the infinite past. Indeed, log-optimum investment policies that at time $n$ look at the $n$-past are sandwiched in asymptotic growth rate between policies that look at only the $k$-past and those that look at the infinite past, and the sandwich closes as $k \rightarrow \infty$.

Let $\{X_t\}$ be a stationary ergodic process with distribution $P$ admitting densities $p(x_0,\ldots, x_{n-1})$ relative to a reference measure $M$ that is finite order Markov with stationary transition kernel. Let $I_M(P)$ denote the relative entropy rate. Then $n^{-1}\log p(X_0,\ldots, X_{n-1}) \rightarrow I_M(P) \mathrm{a.s.} (P).$ We present an elementary proof of the Shannon-McMillan-Breiman theorem and the preceding generalization, obviating the need to verify integrability conditions and also covering the case $I_M(P) = \infty$. A sandwich argument reduces the proof to direct applications of the ergodic theorem.

It is established that if $X$ is a stochastic variable with a normal distribution, then $X^{2n+1}$ has an indeterminate distribution for $n \geq 1$. Furthermore, the distribution of $|X|^\alpha$ is determinate for $0 < \alpha \leq 4$ while indeterminate for $\alpha > 4$.

We consider a time-inhomogeneous regenerative process starting from regeneration at time $s$ and prove, under regularity conditions on the regeneration times, that the distribution of the process in a fixed time interval $\lbrack t, \infty)$ stabilizes as the starting time $s$ tends backward to $-\infty$ (the convergence considered here is in the sense of total variation). This implies the existence of a two-sided time-inhomogeneous process "starting from regeneration at $-\infty$." We also show that if a time-inhomogeneous regenerative process admits a limit law in the traditional forward sense, then it is asymptotically time-homogeneous; thus the backward approach widely extends the class of processes admitting a limit law.