A class of random fields associated with multiple points of a random walk in the plane is studied. It is proved that these fields converge in distribution to analogous fields measuring self-intersections of the planar Brownian motion. The concluding section contains a survey of literature on intersection local times and their renormalizations. A brief look through the first pages of this section could provide the reader with additional motivation for the present work.

Let $T^\varepsilon_k(\lambda; t_1,\ldots,t_k) = \rho(X_{t_1})q^\varepsilon(X_{t_2} - X_{t_1}) \cdots q^\varepsilon(X_{t_k} - X_{t_k - 1}),$ where $X_t$ is a Brownian motion in $\mathbb{R}^2, \lambda(dx) = \rho(x) dx$ and $q^\varepsilon$ converges to Dirac's delta function as $\varepsilon \downarrow 0$. The self-intersection local times of order $k$ are described by a generalized random field $T_k(\lambda; t_1,\ldots,t_k) = \lim_{\varepsilon\downarrow 0} T^\varepsilon_k(\lambda; t_1,\ldots,t_k), \quad\text{for} 0 < t_1 < \cdots < t_k.$ The field "blows up" as $t_i - t_j \rightarrow 0$ for some $i \neq j$. We show that with a proper choice of the coefficients $B^l_k(\varepsilon)$, a generalized random field $\mathscr{J}_k(\lambda; t_1,\ldots,t_k) = \lim_{\varepsilon\downarrow 0}\big\lbrack T^\varepsilon_k(\lambda; t_1,\ldots,t_k) + \sum^{k - 1}_{l = 1}\lbrack B^l_k(\varepsilon)T^\varepsilon_l\rbrack(\lambda; t_1,\ldots,t_k)\big\rbrack$ is well defined for all $0 \leq t_1 \leq \cdots \leq t_k$ and it coincides with $T_k(\lambda; t_1,\ldots,t_k)$ for $t_1 < \cdots < t_k$.

We show that the planar symmetric stable process $X_t$ of index $\frac{4}{3} < \beta < 2$ has an intersection local time $\alpha(x, \cdot)$ which is weakly continuous in $x \neq 0$, while $\alpha(x, \lbrack 0, T\rbrack^2) \sim \frac{c}{|x|^{2 - \beta}}, \quad\text{as} x \rightarrow 0.$

Let $(X, Y)$ solve the martingale problem for a given generator $A$. This paper studies the problem of uniquely characterizing the conditional distribution of $X(t)$ given observations $\{Y(s)\mid 0 \leq s \leq t\}$. We define a filtered martingale problem for $A$ and we show, given appropriate hypotheses on $A$, that the conditional distribution is the unique solution to the filtered martingale problem for $A$. Using these results, we then prove that the solutions to the Kushner-Stratonovich and Zakai equations for filtering Markov processes in additive white noise are unique under fairly general circumstances.

Let $U_1, U_2,\ldots$ be a sequence of independent rv's having the uniform distribution on (0, 1). Let $\hat{F}_n$ be the empirical distribution function based on the transformed uniform spacings $\mathbb{D}_{i, n} := G(nD_{i, n}), i = 1,2,\ldots,n$, where $G$ is the $\exp(1)$ df and $D_{i, n}$ is the $i$th spacing based on $U_1, U_2,\ldots, U_{n - 1}$. In this paper a complete characterization is obtained for the a.s. behaviour of $\lim \sup_{n \rightarrow \infty}b_nV_{n, \nu}$ and $\lim \sup_{n \rightarrow \infty} b_nW_{n, \nu}$ where $\nu \in \lbrack 0, \frac{1}{2}\rbrack, \{b_n\}^\infty_{n = 1}$ is a sequence of norming constants, $V_{n, \nu} = \sup_{0 < t < 1} \frac{n|\hat{F}_n(t) - t|}{t^{1 - \nu}} \quad\text{and}\quad W_{n, \nu} = \sup_{0 < t < 1} \frac{n|\hat{F}_n(t) - t|}{(1 - t)^{1 - \nu}}.$ It turns out that compared with the i.i.d. case only $W_{n, \nu}$ behaves differently. The results imply, e.g., laws of the iterated logarithm for $\log(n^{\nu - 1}V_{n, \nu})$ and $\log(n^{\nu - 1}W_{n, \nu})$. Of independent interest is the theorem on the lower-upper class behaviour of the maximal spacing, which gives the final solution for this problem and generalizes some recent results in the literature.

Characterizations of laws of the iterated logarithm for the supremum of weighted uniform $\lbrack 0, 1\rbrack^d$ empirical processes taken over increasingly smaller regions near the origin are obtained. These results have proven to be a valuable tool in the derivation of laws of the iterated logarithm for sums of extreme values. They also constitute a further continuation of the study of the almost sure behavior of weighted uniform empirical processes, which in a certain sense was begun by Csaki.

The asymptotic distributions of $L_p$ functionals of weighted uniform quantile and empirical processes are studied. The asymptotic laws obtained are represented in terms of Gaussian and Poisson integrals.

Let $U_1, U_2,\ldots$ be a sequence of independent uniform (0, 1) random variables, and for $1 \leq k \leq n$ let $\xi_p(n, k)$ denote the $p$th quantile, $0 < p < 1$, corresponding to the block $U_{n -k + 1},\ldots,U_n$. In this paper we investigate the a.s. limiting behavior of $\xi_p(n, a_n)$ when $a_n$ is an integer sequence, $1 \leq a_n \leq n$, and $\lim_{n \rightarrow \infty}a_n/\log n = \beta \in \lbrack 0, \infty\rbrack$. In addition, we investigate the a.s. limiting behavior of $\max_{a_n \leq k \leq n}\xi_p(n, k)$ and other maxima involving the $\xi_p(n, k)$'s.

Let $(g_i)_{i \geq 1}$ be an i.i.d. sequence of standard normal r.v.'s. Let $A$ be a family of sequences $a = (a_i)_{i \geq 1}, a_i \geq 0$. We relate the quantity $E \operatorname{Sup}_{a \in A}\sum_{i \geq 1}a_i|g_i|$ and the geometry of $A$.

We present a simple proof with sharp estimates of Dvoretzky's theorem on the existence of almost spherical sections having large dimension in arbitrary convex bodies in $R^N$.

Bounds are given on the mean time taken by a strong Markov process to visit all of a finite collection of subsets of its state space. These bounds are specialized to Brownian motion on the surface of the unit sphere $\Sigma_p$ in $R^p$. This leads to bounds on the mean time taken by this Brownian motion to come within a distance $\varepsilon$ of every point on the sphere and bounds on the mean time taken to come within $\varepsilon$ of every point or its opposite. The second case is related to the Grand Tour, a technique of multivariate data analysis that involves a search of low-dimensional projections. In both cases the bounds are asymptotically tight as $\varepsilon \rightarrow 0$ on $\Sigma_p$ for $p \geq 4$.

Let $(X_t)_{t \geq 0}$ denote the measure-valued critical branching Brownian motion. When the support of the initial state, $X_0$, is bounded, temporally global results are given concerning the range, i.e., the size of the supports of $(X_t)_{t \geq 0}$, and the hitting (i.e., charging) probabilities of distant balls are evaluated asymptotically; they depend strongly on the dimension, $d$, of the underlying Euclidean space $\mathbb{R}^d$. In contrast, in the case $d = 1$ and $X_0 = \lambda$ (Lebesgue measure), it is shown that (spatially) local extinction occurs. Also extensions are indicated for the case of an infinite variance branching mechanism; these results are also dimensionally dependent.

Kolmogorov's extension theorem is generalized so that the time domain is a random set. This is applied to the construction of stochastic processes with random time domains, generalizing certain results of Dynkin and Kuznetsov.

Two extensions of the Ito integral are developed, and put in the perspective of derivative operators in the Malliavin calculus. The divergence operator, $\delta$, is constructed, and its properties and action on these two extended integrals are described. Discussion of iterated stochastic integrals and the extended stochastic integrals as functions of their upper limits is also included.

If $X^{(n)}$ is a sequence of semimartingales, converging to a semimartingale $X$, and such that $\lbrack X^{(n)}, X^{(n)}\rbrack$ converges to $\lbrack X, X\rbrack$, then all higher-order variations and all the iterated integrals of $X^{(n)}$ converge jointly to the respective functionals of $X$.

Let $F$ be a distribution function on $\lbrack 0, 1\rbrack^d$, and let $W_F$ be the Gaussian process that is the weak limit of the empirical process determined by $F$. If $G$ is a function on $\lbrack 0, 1\rbrack^d$, upper and lower bounds are found for $P(\sup_{t \in \lbrack 0, 1\rbrack^d}|W_F(t) - G(t)| \leq \varepsilon)$.

A rigorous definition of two-parameter point processes is given as a distribution of a denumerable number of random points in the plane. A characterization with stopping lines and relation with predictability are obtained. Using the one-parameter multivariate point-process representation, a general representation theorem for a wide class of martingales is presented, which extends the representation theorem with respect to a Poisson process.

Heyde and Brown (1970) established a bound on the rate of convergence in the central limit theorem for discrete time martingales having finite moments of order $2 + 2\delta$ with $0 < \delta \leq 1$. In the present paper a modification of the methods developed by Bolthausen (1982) is applied to show the validity of this result for all $\delta > 0$. Moreover, an example is constructed demonstrating that this bound is asymptotically exact for all $\delta > 0$. The result for discrete time martingales is then used to derive the corresponding bound on the rate of convergence in the central limit theorem for locally square integrable martingales with continuous time.

We use Poincare-type inequalities to prove the sufficiency and necessity of the Lindeberg condition in the central limit theorem.

If the means and variances of a sequence of random variables converge, and all semiinvariants (cumulants) of sufficiently high order tend to zero, then the variables converge in distribution to a normal distribution. Thus no information is needed on the remaining (finitely many) semiinvariants. This is applied to give a new criterion for asymptotic normality of sums of dependent variables. An example is included where this criterion is applied to the number of induced subgraphs of a particular type in a random graph.

A central limit theorem is proved for some strictly stationary $\rho$-mixing sequences with infinite second moments. The condition on the tails of the marginal distribution is the same as in the corresponding classic result for i.i.d. sequences. The mixing rate is essentially the slowest possible for this result.

Rootzen (1978) gives a sufficient condition for sample continuity of moving average processes with respect to stable motion with index $\alpha$ less than two. We provide a simple proof of this criterion for $\alpha < 1$ and show that the condition is then also necessary for continuity of the process. The same result holds for the moving-maximum process. A description of the local behaviour of the sample functions of such processes is given.

Consider a $\{0, 1\}$-valued strictly stationary stochastic process $\{X_1,X_2,\ldots\}$. Let $k$ and $l$ be natural numbers and define $y_i = 0$ or 1 according as $x_1 + \cdots + x_{i+k-1}$ is even or odd. Then, for $1 \leq j \leq l$ set $S_j(x_1 \cdots x_n) = \sum_{0\leq i \leq m - 1}y_{j+il}$. We consider all processes that have $(S_1,\ldots,S_l)$ as sufficient statistics. We obtain explicit formulas for the distributions of the processes that are extreme points. We also represent these processes as finitary processes and use this representation to investigate their pairwise independence, ergodicity and mixing properties.

Let $w$ be the spectral density function of a weakly stationary stochastic process with $w = |h|^2, h$ being an outer function in the upper half plane, and let $\rho^\ast(a) = \operatorname{dist}(e^{ita}h/\bar{h}, H^\infty)$, where $H^\infty$ is the space of boundary functions on $R$ for bounded analytic functions in the upper half plane. It is shown that the standard deviation of the difference between the infinite predictor and the finite predictor from the past of length $T$ does not exceed $\rho^\ast(T)/(1 - \rho^\ast(T))$ times the prediction error of the infinite predictor. Some other estimates relating to the difference between the infinite predictor and the finite predictor are also discussed.

Let $(X_t)$ be a diffusion process on the interval $(r_1, r_2)$ where $r_2$ is inaccessible. For $r_1 < z < r_2$, let $T_z$ be the first passage time to $z$, and define $L_u = \int_{\{t: 0 \leq t \leq T_z, X_t > u\}} J(X_t) dt$, where $J$ is a particular function determined by the generator of the diffusion. An explicit asymptotic expression is obtained for the probability $P(L_u > y|X_0 = x)$, for $u \rightarrow r_2$, fixed $y > 0$, and $r_1 < x < r_2$. From this the corresponding asymptotic form of the distribution of the sojourn time, $mes(t: 0 \leq t \leq T_z, X_t > u)$ is determined when $r_2 = \infty$. Related theorems are given for the distribution of $L_u - L_v$, for $u, v \rightarrow \infty, u \leq v$ and $u/v \rightarrow 1$. Finally, the results are extended to the long-term sojourn integral $L^\ast_u = \int_{\{t: 0 \leq t \leq S(u), X_t > u\}} J(X_t) dt$, where $S(u) \rightarrow \infty$ for $u \rightarrow r_2$.

Motivated by several classical sequential decision problems, we study herein the following type of boundary crossing problems for certain nonlinear functions of sample means. Let $X_1, X_2,\ldots$ be i.i.d. random vectors whose common density belongs to the $k$-dimensional exponential family $h_\theta(x) = \exp\{\theta'x - \psi(\theta)\}$ with respect to some nondegenerate measure $\nu$. Let $\bar{X}_n = (X_1 + \cdots + X_n)/n, \hat\theta_n = (\nabla\psi)^{-1}(\bar{X}_n)$, and let $I(\theta, \lambda) = E_\theta\log\{h_\theta(X_1)/h_\lambda(X_1)\}$ ( = Kullback-Leibler information number). Consider stopping times of the form $T_c(\lambda) = \inf\{n: I(\hat\theta_n, \lambda) \geq n^{-1}g(cn)\}, c > 0$, where $g$ is a positive function such that $g(t) \sim \alpha \log t^{-1}$ as $t \rightarrow 0$. We obtain asymptotic approximations to the moments $E_\theta T^r_c(\lambda)$ as $c \rightarrow 0$ that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$. We also study the probability that $\bar{X}_{Tc(\lambda)}$ lies in certain cones with vertex $\nabla\psi (\lambda)$. In particular, in the one-dimensional case with $\lambda > \theta$, we consider boundary crossing probabilities of the form $P_\theta\{\hat\theta_n \geq \lambda \text{and} I(\hat\theta_n, \lambda) \geq n^{-1} g(cn) \text{for some} n\}$. Asymptotic approximations (as $c \rightarrow 0$) to these boundary crossing probabilities are obtained that are uniform in $\theta$ and $\lambda$ with $|\lambda - \theta|^2/c \rightarrow \infty$.

It is shown that if a discrete-time Markov chain on the state space $\{0, 1,\ldots\}$ has a transition probability matrix $\mathbf{P}$ and a transition survival probability matrix $\mathbf{Q}$ which is totally positive of order two $(TP_2)$, where $Q(i, j) = \sum_{k\geq j}P(i, k)$, then the first-passage time from state 1 to state 0 has decreasing failure rate (DFR). This result is used to show that (i) the sum of a geometrically distributed number (i.e., geometric compound) of i.i.d. DFR random variables is DFR, and (ii) the number of customers served during a busy period in an M/G/1 queue with increasing failure-rate service times is DFR. Recent results of Szekli (1986) and the closure property of i.i.d. DFR random variables under geometric compounding are combined to show that the stationary waiting time in a GI/G/1 (M/G/1) queue with DFR (increasing mean residual life) service times is DFR. We also provide sufficient conditions on the inter-renewal times under which the renewal function is concave. These results shed some light on a conjecture of Brown (1981).

A real-valued function $g$ of two vector arguments $\mathbf{x}$ and $\mathbf{y} \in R^n$ is said to be arrangement increasing if it increases in value as the arrangement of components in $\mathbf{x}$ becomes increasingly similar to the arrangement of components in $\mathbf{y}$. Hollander, Proschan and Sethuraman (1977) show that the convolution of arrangement increasing functions is arrangement increasing. This result is used to generate some interesting probability inequalities of a geometric nature for exchangeable random vectors. Other geometric inequalities for families of arrangement increasing multivariate densities are also given, and some moment inequalities are obtained.

Tight bounds on the sup metric between the exponential distribution and new better (worse) than used in expectation (NBUE, NWUE) distributions are established in terms of the proximity of the second moments of the distributions concerned. Real variable methods are used to identify the extremal distributions that attain the bounds. Similar methods establish similar results for the harmonic NBUE and NWUE classes of distributions.