A series of observations $\{\xi_1, \xi_2, \xi_3,\ldots\}$ is presented to us and at each time $n$, when we have observed the first $n$ of them, we are called upon to give our guess for what stochastic process produced the data. A universal scheme is given which, for any Bernoulli process (not necessarily independent), gives a sequence of processes that converges in a strong sense (the $\bar{d}$-metric) to the real process. In addition to this main result, many others are given which put it into proper perspective. In particular it is shown that in a certain sense the class of Bernoulli processes is the largest one for which such a universal scheme is possible.

There is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree. It equals the exponential of the Hausdorff dimension of the boundary in an appropriate metric. Its importance for probabilistic processes on a tree is shown in several ways, including random walk and percolation, where it provides points of phase transition.

With each random walk on $\mathbb{Z}^2$ we associate a functional related to the number of steps which the walk spends in sites occupied at least $k$ times. We show that if our random walk is in the domain of attraction of a stable process of order greater than $2(2k - 2)/(2k - 1)$, then our functional coverges to the intersection local time of the limiting process.

A new proof of the nonincrease of Brownian paths is given.

Let $X$ and $Y$ be independent two-dimensional Brownian motions, $X(0) = (0, 0), Y(0) = (\varepsilon, 0)$, and let $p(\varepsilon) = P(X\lbrack 0, 1 \rbrack \cap Y\lbrack 0, 1 \rbrack = \varnothing), q(\varepsilon) = \{Y\lbrack 0, 1 \rbrack \text{does not contain a closed loop around} 0\}$. Asymptotic estimates (when $\varepsilon \rightarrow 0$) of $p(\varepsilon), q(\varepsilon)$, and some related probabilities, are given. Let $F$ be the boundary of the unbounded connected component of $\mathbb{R}^2\backslash Z\lbrack 0, 1 \rbrack$, where $Z(t) = X(t) - tX(1)$ for $t \in \lbrack 0, 1 \rbrack$. Then $F$ is a closed Jordan arc and the Hausdorff dimension of $F$ is less or equal to $3/2 - 1/(4\pi^2)$.

Consider two Brownian motions $B^1_{s_1}$ and $B^2_{s_2}$, each taking values on an interval $\lbrack 0, a_i \rbrack, i = 1, 2$, with absorption at the endpoints. The time evolution of the two processes can be controlled separately: i.e., we can alternate between letting $B^1_{s_1}$ run while freezing $B^2_{s_2}$ and letting $B^2_{s_2}$ run while freezing $B^1_{s_1}$. This results in a switched process that evolves in the rectangle, $D = \lbrack 0, a_1 \rbrack \times \lbrack 0, a_2 \rbrack$ like a horizontal Brownian motion when $B^2_{s_2}$ freezes and like a vertical Brownian motion when $B^1_{s_1}$ freezes. Let $f(x_1, x_2)$ be a nonnegative continuous payoff function defined on the boundary $\partial D$ of $D$. A control consists of a switching strategy and a stopping time $\tau$. We study the problem of finding an optimal control which maximizes the expected payoff obtained at time $\tau$ (stopping in the interior results in zero reward). In the interior of the rectangle, the optimal switching strategy is determined by a partition into three sets: a horizontal control set, a vertical control set and an indifference set. We give an explicit characterization of these sets in the case when the payoff function is either linear or strongly concave on each face.

Let $T_\theta$ be the first exit time of a symmetric stable process [with parameter $\alpha \in (0, 2)$] from a wedge of angle $2\theta, 0 < \theta < \pi$. Then there are constants $p_{\theta, \alpha} > 0$ such that for starting points $x$ in the wedge, $E_xT^p_\theta < \infty$ if $0 < p < p_{\theta, \alpha}$ and $E_xT^p_\theta = \infty$ if $p > p_{\theta, \alpha}$. We characterize $p_{\alpha, \theta}$ and obtain upper and lower bounds.

Let $X_1, X_2,\ldots$ be a Markov process with state space $E$, a Polish space. Let $L_n(\omega, A) = n^{-1}\sum^{n - 1}_{j = 0}1_A(X_j(\omega))$ denote the normalized occupation time measure. If $\mu$ is a probability measure on $E, G$ is a weak neighborhood of $\mu$, and if $V \subset E$, then we obtain asymptotic lower bounds for probabilities $P^x\lbrack L_n(\omega, \cdot) \in G, X_j(\omega) \in V, 0 \leq j \leq n - 1 \rbrack$ in terms of $I(\mu)$, the rate function of Donsker and Varadhan. Our assumptions are weaker than those imposed by Donsker and Varadhan, and the proof works without any essential change in the continuous time case as well. In fact, the same proofs apply to certain bounded additive functionals: Let $r \geq 0$ and let $f: \Omega \rightarrow \mathbf{B}$ be bounded $\mathscr{F}^0_r$-measurable, where $\Omega$ is the sample space with the product topology (Skorohod topology in the continuous time case) and $\mathbf{B}$ is a separable Banach space; let $\theta_k: \Omega \rightarrow \Omega$ be the shift operator, i.e., $\theta_k\omega(j) = \omega(k + j)$. Then we get lower bounds for probabilities involving $n^{-1}(f(\omega) + f(\theta_1\omega) + \cdots + f(\theta_{n - 1}\omega))$ in place of $L_n(\omega, \cdot)$. In this latter situation, the rate function has to be the entropy function $H(Q)$ of Donsker and Varadhan.

Let $\nu$ denote the value function of a partially observed control problem. If $\nu$ is once differentiable in a certain direction $\hat{B}$, then optimal controls are characterized by a feedback involving the directional derivative $\hat{B}\nu$. It is also shown that $\nu$ satisfies the corresponding Bellman equation, an infinite-dimensional PDE on the space of measures, in the viscosity sense of Crandall and Lions.

Let $\{X_t, t \geq 0\}$ be an $R^d$-valued, symmetric, right Markov process with stationary transition density. Let $\{\hat{X}_t, t \geq 0\}$ denote the version of $X_t$ "killed" at an exponential random time, independent of $X_t$. Associated with $\hat{X}_t$ is a Green's function $g(x, y)$, which we assume satisfies $0 < g(x, x) < \infty$ for all $x$ and a local time $\{L_x, x \in R^d\}$. It follows from an isomorphism theorem of Dynkin that $L_x$ has continuous sample paths whenever $\{G(x), x \in R^d\}$, a Gaussian process with covariance $g(x, y)$, does. In this paper we use Dynkin's theorem to show that $L_x$ satisfies the central limit theorem in the space of continuous functions on $R^d$ if and only if $G(x)$ has continuous sample paths. This result strengthens a result of Adler and Epstein on the construction of the free field by means of a central limit theorem involving the local time, in the case when the local time is a point indexed process. In order to apply Dynkin's theorem the following result is obtained: The square of a continuous Gaussian process satisfies the central limit theorem in the space of continuous functions.

We consider the large values of a locally stationary Gaussian process which satisfies Berman's condition on the long range dependence. The paper presents some limit results on the exceedances of the process above a certain general smooth high boundary. This allows deriving the limiting distribution of the maximum up to time $T$, for example, in the case of a standardized process with a constant boundary or in the case of a nonstandardized process with a smooth trend.

Given a stationary Gaussian vector process $(X_m, Y_m), m \in Z$, and two real functions $H(x)$ and $K(x)$, we define $Z^n_H = A^{-1}_n\sum^{n - 1}_{m = 0} H(X_m)$ and $Z^n_K = B^{-1}_n\sum^{n - 1}_{m = 0} K(Y_m)$, where $A_n$ and $B_n$ are some appropriate constants. The joint limiting distribution of $(Z^n_H, Z^n_K)$ is investigated. It is shown that $Z^n_H$ and $Z^n_K$ are asymptotically independent in various cases. The application of this to the limiting distribution for a certain class of nonlinear infinite-coordinated functions of a Gaussian process is also discussed.

Let $\{v_{ij}\}, i, j = 1, 2, \ldots,$ be i.i.d. symmetric random variables with $\mathbb{E}(\nu^4_{11}) < \infty$, and for each $n$ let $M_n = (1/s)V_n V^T_n$, where $V_n = (v_{ij}), i = 1, 2, \ldots, n, j = 1, 2, \ldots, s = s(n)$ and $n/s \rightarrow y > 0$ as $n \rightarrow \infty$. Denote by $O_n \Lambda_n O^T_n$ the spectral decomposition of $M_n$. Define $X \in D\lbrack 0, 1 \rbrack$ by $X_n(t) = \sqrt{n/2} \sum^{\lbrack nt \rbrack}_{i = 1}(y^2_i - 1/n)$ where $(y_1, y_2, \ldots, y_n)^T = O^T(\pm 1/\sqrt{n}, \pm 1/ \sqrt{n}, \ldots, \pm 1/\sqrt{n})^T$. It is shown that $X_n \rightarrow_\mathscr{D} W^0$ as $n \rightarrow \infty$, where $W^0$ is a Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of $M_n$ for $n$ large and for general $v_{11}$.

A Dirichlet form on a large (complicated or multidimensional) space may be carried over onto a small (simple or one-dimensional) space. Here conditions are given ensuring the induced form is regular. A capacitary inequality between the two forms allows one to estimate the probability of a large deviation on the large space by that on the small space. Also asymptotically sharp results are derived in a one-dimensional setting.

A way of transforming a multiparameter point process into a Poisson process is given. As in the one-parameter case, the compensator characterization of the Poisson process plays an important role in random space change for the multiparameter point process. We use the characterization of planar Poisson processes in terms of the 1-compensator by Brown to derive random space-change theorems. The results obtained hold under weaker conditions than those in Merzbach and Nualart.

A tandem queue with $m$ single server stations and unlimited interstage storage is considered. Such a tandem queue is described by a generic sequence of nonnegative random vectors in $R^{m + 1}$. The first $m$ coordinates of the $k$th element of the generic sequence represent the service times of the $k$th unit in $m$ single server queues, respectively, and the $(m + 1)$th coordinate represents the interarrival time between the $k$th and $(k + 1)$th units to the tandem queue. The sequences of vectors $\tilde{w}_k = (w_k(1), w_k(2),\ldots, w_k(m))$ and $\tilde{W}_k = (W_k(1), W_k(2),\ldots, W_k(m))$, where $w_k(i)$ represents the waiting time of the $k$th unit in the $i$th queue and $W_k(i)$ represents the sojourn time of the $k$th unit in the first $i$ queues, are studied. It is shown that if the generic sequence is asymptotically stationary in some sense and it satisfies some natural conditions then $\mathbf{w} = \{\tilde{w}_k, k \geq 1\}$ and $\mathbf{W} = \{\tilde{W}_k, k \geq 1\}$ are asymptotically stationary in the same sense. Moreover, their stationary representations are given and the heavy traffic approximation of that stationary representation is given.

The object of study in this paper is a Brownian model of a multiclass service station. Such Brownian models arise as heavy traffic limits of conventional queueing models in which several different types or classes of customers are processed through a common service facility. Assuming that the Brownian service station is initialized with its stationary distribution, four different model characteristics are shown to be equivalent, and the station is said to be quasireversible if these equivalent conditions pertain. Three of the four conditions characterize the vector departure process from the Brownian service station, and our definition of quasireversibility parallels that proposed by F. P. Kelly for conventional queueing models. The last of our four conditions is expressed directly in terms of primitive model parameters, so one may easily determine from basic data whether or not a Brownian station model is quasireversible. Rather than characterizing the complete vector of departure processes from a Brownian service station, we prove a more general theorem expressed in terms of arbitrary linear combinations of the departure processes; this yields a generalized notion of quasireversibility that will play an important role in future work. To be more specific, in a future paper on multiclass Brownian network models, it will be shown that there is an intimate relationship between product form stationary distributions and the generalized notion of quasireversibility developed here.

Let $\hat{F}_n$ denote the empirical distribution function for a sample of $n$ i.i.d. random variables with distribution function $F$. In 1956 Dvoretzky, Kiefer and Wolfowitz proved that $P\big(\sqrt{n} \sup_x(\hat{F}_n(x) - F(x)) > \lambda\big) \leq C \exp(-2\lambda^2),$ where $C$ is some unspecified constant. We show that $C$ can be taken as 1 (as conjectured by Birnbaum and McCarty in 1958), provided that $\exp(-2\lambda^2) \leq \frac{1}{2}$. In particular, the two-sided inequality $P\big(\sqrt{n} \sup_x|\hat{F}_n(x) - F(x)| > \lambda\big) \leq 2 \exp(-2\lambda^2)$ holds without any restriction on $\lambda$. In the one-sided as well as in the two-sided case, the constants cannot be further improved.

Empirical versions of appropriate centering and scale constants for random variables which can fail to have second or even first moments are obtainable in various ways via suitable modifications of the summands in the partial sum. This paper discusses a particular modification, called censoring (which is a kind of winsorization), where the (random) number of summands altered tends to infinity but the proportion altered tends to zero as the number of summands increases. Some analytic advantages inherent in this approach allow a fairly complete probabilistic and empirical theory to be developed, the latter involving the study of studentized or self-normalized sums. In particular, the joint asymptotic distributions of the empirically censored quantities of center and scale are determined as well as precise criteria for convergence to each of the allowable limit laws. Applications to the Feller class and domains of attraction are also considered.

We establish necessary and sufficient conditions for convergence of the distribution function of a bootstrapped mean, suitably normalized. It turns out that for convergence to occur, the sampling distribution must either be in the domain of attraction of the normal distribution or have slowly varying tails. In the first case the limit is normal; in the latter, Poisson. Between these two extremes of light tails and extremely heavy tails, the bootstrap distribution function of the mean does not converge in probability to a nondegenerate limit. However, it may converge in distribution. We show that when there is no convergence in probability, a small number of extreme sample values determine behaviour of the bootstrap distribution function. This result is developed and used to interpret recent work of Athreya.

The asymptotic normality of linear serial rank statistics introduced by Hallin, Ingenbleek and Puri (1985) for the problem of testing white noise against ARMA alternatives is established for $\varphi$-mixing as well as strong mixing sequences of random variables. Applications in Markov processes and ARMA processes in time series are provided.

Define $Y_0$ to be a geometric convolution of $X$ if $Y_0$ is the sum of $N_0$ i.i.d. random variables distributed as $X$, where $N_0$ is geometrically distributed and independent of $X$. It is known that if $X$ is nonnegative with finite second moment, then as $p \rightarrow 0, Y_0/EY_0$ converges in distribution to an exponential distribution with mean 1. We derive an upper bound for $d(Y_0)$, the sup norm distance between $Y_0$ and an exponential with mean $EY_0$. This upper bound is $d(Y_0) \leq \operatorname{cp}$ for $0 < p \leq \frac{1}{2}$, where $c = EX^2/(EX)^2$. It is asymptotically $(p \rightarrow 0)$ tight. Also derived is a bound for $d(Y_0 + Z)$, where $Z$ is independent of $Y_0$.

There are many known asymptotic estimates of the number of zeros of the polynomial $T(\theta) = g_1 \cos \theta + g_2 \cos 2\theta + \cdots + g_n \cos n \theta$ for $n \rightarrow \infty$, where $g_i (i = 1, 2,\ldots, n)$ is a sequence of independent normally distributed random variables with mathematical expectation 0 and variance 1. The present paper provides an estimate of the expected number of times that such a polynomial assumes the real value $K$. It is shown that the results for $K = 0$ are valid when $K = o(\sqrt{n})$.

This paper shows that Dykstra's procedure for finding the $I$-projection onto the intersection of closed convex sets holds in general. It does this by first showing that each of the $I$-projections onto individual convex sets defined in Dykstra's iterative procedure exists and that no condition such as imposed by Dykstra is required to prove the convergence of the iterative procedure to a unique $I$-projection.