If $(B_t)_{t\geq 0}$ is Brownian motion on $\mathbb{R},$ if $A(t, x) \equiv \int^t_0I_{\{B_s\leq x\}} ds$ and if $\tau(\cdot, x)$ is the right-continuous inverse to $A(\cdot, x)$, then the process $\tilde{B}(t, x) \equiv B(\tau(t, x))$ is a reflecting Brownian motion in $(-\infty, x\rbrack$. If $\mathscr{E}_x$ denotes the $\sigma$-field generated by $\tilde{B}(\cdot, x)$, then $(\mathscr{E}_x)_{x\in\mathbb{R}}$ forms a filtration. It has been proved recently that all $(\mathscr{E}_x)$-martingales are continuous, in common with the martingales on the Brownian filtration. Here we shall prove that, as with the Brownian filtration, all $(\mathscr{E}_x)$-martingales can be written as stochastic area integrals with respect to local time. This requires a theory of such integrals to be developed; the first version of this was given by Walsh some years ago, but we consider the account presented here to be definitive. We apply this theory to an investigation of stochastic line integrals of local time along curves which need not be adapted processes and illustrate these constructs by identifying the compensator of the supermartingale $(L(\tau(t, x), x))_{x\geq a}$ previously studied by McGill.

Ustunel and Zakai have recently obtained a necessary and sufficient condition for two multiple Wiener integrals with respect to the same Brownian motion to be independent. In the present note, the sufficiency of their condition is shown to be a simple consequence of the classical relationship between multiple Wiener integrals and Hermite polynomials. The original proofs use Malliavin calculus.

Bounds are found on the transition densities and Green functions for Brownian motion with normal reflection in Holder and Lipschitz domains. For Lipschitz domains, reflecting Brownian motion and boundary local time are constructed, a Harnack inequality valid up to the boundary is proved, a probabilistic solution to the Neumann problem is given and the Kuramochi boundary is identified.

We consider a family of periodic diffusion processes in $R^n$ with homogenization and a small parameter multiplying the diffusion coefficient. We establish a large deviations principle and as an application we derive an iterated logarithm law for periodic diffusions.

We show the well-posedness for the martingale problem associated with a strictly elliptic operator with coefficients bounded and continuous on $\mathbb{R}^n$ except for a countable set having at most one cluster point.

In this paper we study the transition density $P_t(x, y)$ of a nondegenerate diffusion process by using the stochastic control method invented by Fleming and the idea of stochastic parallel translation. We obtain a two-sided estimate for $P_t(x, y)$ as well as some bounds for the derivatives of $\log P_t(x, y)$.

The symmetry groups of the potential theory of a Markov process $X_t$ are used to introduce new algebraic and topological structures on the state space and the process. For example, let $G$ be the collection of bijections $\varphi$ on $E$ which preserve the collection of excessive functions. Assume there is a transitive subgroup $H$ of the symmetry group $G$ such that the only map $\varphi \in H$ fixing a point $e \in E$ is the identity map on $E$. There is a bijection $\Psi: E \rightarrow H$ so that the algebraic structure of $H$ can be carried to $E$, making $E$ into a group. If there is a left quasi-invariant measure on $E$, then there is a topology on $E$ making $E$ into a locally compact second countable metric group. There is also a time change $\tau(t)$ of $X_t$ such that $X_{\tau(t)}$ is a translation invariant process on $E$ and $X_{\tau(t)}$ is right-continuous with left limits in the new topology.

A functional central limit theorem is established for Markov chains in random environments under the assumption of existence of a finite invariant, ergodic measure and a mixing condition. These conditions are always satisfied when the state space is finite.

Let $P = (p_{ij}), i, j = 1,2,\ldots, n$ be the matrix of a recurrent Markov chain with stationary vector $\nu > 0$ and let $R = (r_{ij}), i, j = 1,2,\ldots, n$ be a matrix, where $r_{ij} = v_ip_{ij}$. If $R$ is a symmetric matrix, we improve Alpern's rotational representation of $P$. By this representation we characterize the reversible Markov chains.

In this paper, we study spin systems, probabilistic cellular automata and interacting particle systems, which are Markov processes with state space $\{0, 1\}^{\mathbf{Z}^n}$. Restricting ourselves to attractive systems, we consider the stationary processes obtained when either of two distinguished stationary distributions is used, the smallest and largest stationary distributions with respect to a natural partial order on measures. In discrete time, we show that these stationary processes with state space $\{0, 1\}^{\mathbf{Z}^n}$ and index set $\mathbf{Z}$ are isomorphic (in the sense of ergodic theory) to an independent process indexed by $\mathbf{Z}$. In the translation invariant case, we prove the stronger fact that these stationary processes, viewed as $\{0, 1\}$-valued processes with index set $\mathbf{Z}^n \times \mathbf{Z}$ (space-time), are isomorphic to an independent process also indexed by $\mathbf{Z}^n \times \mathbf{Z}$. Such processes are called Bernoulli shifts. Finally, we extend all of these results to continuous time.

A sufficient condition is given for a sequence of partial-sum set-indexed processes with nonuniform $\phi$-mixing condition to converge to Brownian motion. The main result (Theorem 1.1) is an extension of the similar results of Goldie and Greenwood by weakening the $\phi$-mixing condition. An application (Corollary 4.2) to certain Gibbs fields is given.

Limit theorems are proved for the range of $d$-dimensional random walks in the domain of attraction of a stable process of index $\beta$. The range $R_n$ is the number of distinct sites of $\mathbb{Z}^d$ visited by the random walk before time $n$. Our results depend on the value of the ratio $\beta/d$. The most interesting results are obtained for $2/3 < \beta/d \leq 1$. The law of large numbers then holds for $R_n$, that is, the sequence $R_n/E(R_n)$ converges toward some constant and we prove the convergence in distribution of the sequence $(\operatorname{var} R_n)^{-1/2}(R_n - E(R_n))$ toward a renormalized self-intersection local time of the limiting stable process. For $\beta/d \leq 2/3$, a central limit theorem is also shown to hold for $R_n$, but, in contrast with the previous case, the limiting law is normal. When $\beta/d > 1$, which can only occur if $d = 1$, we prove the convergence in distribution of $R_n/E(R_n)$ toward some constant times the Lebesgue measure of the range of the limiting stable process. Some of our results require regularity assumptions on the characteristic function of $X$.

A random vector is $\min$-stable (or jointly negative exponential) if any weighted minimum of its components has a negative exponential distribution. The vectors can be subordinated to a two-dimensional homogeneous Poisson point process through positive $\mathscr{L}_1$ functions called spectral functions. A critical feature of this representation is the point of the Poisson process, called the location, that defines a $\min$-stable random variable. A measure of association between $\min$-stable random variables is used to define mixing conditions for $\min$-stable processes. The association between two $\min$-stable random variables $X_1$ and $X_2$ is defined as the probability that they share the same location and is denoted by $q(X_1, X_2).$ Mixing criteria for a $\min$-stable process $X(t)$ are defined by how fast the association between $X(t)$ and $X(t + s)$ goes to zero as $s \rightarrow \infty$. For some stationary processes (including the moving-minimum process), conditions on the spectral functions are derived in order that the processes satisfy mixing conditions.

Berry-Esseen theorems are proved in the multidimensional central limit theorem without using Fourier methods. An effective and simple estimate of the error in the CLT for sums and convex sets using Stein's method and induction is derived. Furthermore, the error in the CLT for multivariate functions of independent random elements is estimated extending results of van Zwet and Friedrich to the multivariate case.

Laws of the iterated logarithm are derived for row sums of triangular arrays of independent random variables, in the context of nonparametric regression estimators. These laws provide exact strong convergence rates for kernel type nonparametric regression estimators. They apply to the important case where design points are conditioned upon, and permit the design to be multivariate. We impose minimal conditions on the error distribution; in fact, only finite variance is needed.

It is proved that, for classes of functions $\mathscr{F}$ satisfying some measurability, the empirical processes indexed by $\mathscr{F}$ and based on $P \in \mathscr{P}(S)$ satisfy the central limit theorem uniformly in $P \in \mathscr{P}(S)$ if and only if the $P$-Brownian bridges $G_p$ indexed by $\mathscr{F}$ are sample bounded and $\rho_p$ uniformly continuous uniformly in $P \in \mathscr{P}(S)$. Uniform exponential bounds for empirical processes indexed by universal bounded Donsker and uniform Donsker classes of functions are also obtained.

Let $X_{1,n}\leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent random variables with a common distribution function $F$ and let $k_n$ be positive integers such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow \alpha$ as $n \rightarrow \infty$, where $0 \leq \alpha < 1$. We find necessary and sufficient conditions for the existence of normalizing and centering constants $A_n > 0$ and $C_n$ such that the sequence $E_n = \frac{1}{A_n}\bigg\{\sum^{k_n}_{i=1} X_{n+1-i,n} - C_n\bigg\}$ converges in distribution along subsequences of the integers $\{n\}$ to nondegenerate limits and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $C_n$ such that $E_n$ be asymptotically normal along a given subsequence, and with suitable $A_n$ and $C_n$ determine the limiting distributions of $E_n$ along the whole sequence $\{n\}$ when $F$ is in the domain of attraction of an extreme value distribution.

We introduce a class of so-called conditional $U$-statistics, which generalize the Nadaraya-Watson estimate of a regression function in the same way as Hoeffding's classical $U$-statistics is a generalization of the sample mean. Asymptotic normality and weak and strong consistency are proved.

A huge body of if and only if theorems can be obtained based on certain strong embedding theorems for the partial sum process $\mathbb{S}_n$ and the uniform empirical and quantile processes $\mathbb{U}_n$ and $\mathbb{V}_n$. This embedding was accomplished in 1986 by M. Csorgo, S. Csorgo, L. Horvath and D. Mason. Their embedding is beautifully formulated so that many necessary and sufficient type results can be established using it. It is worthwhile to have an accessible proof. Indeed, these authors have since produced two papers that obtain the essential form of their result without appealing to earlier work of J. Komlos, P. Major and G. Tusnady. Indeed, this present paper does this for the finite sampling process $\mathbb{R}_n$ and the weighted empirical process $\mathbb{W}_n$. These latter results are entirely new and are the main objectives of the present paper. The applications of these latter results will appear elsewhere.

We develop the linear least squares prediction theory for some classes of nonstationary processes having a Fourier spectral representation. We study time domain as well as spectral domain properties for these processes, such as a Wold decomposition and a decomposition for matrix bimeasures. We also obtain an autoregressive representation for the optimum predictor.

Let $m (\geq 2)$ be a positive integer; $I_m$ be the $m \times m$ identity matrix; and $\Sigma$ and $A$ be symmetric $m \times m$ matrices, where $\Sigma$ is positive definite. By proving that the function $\phi_\alpha(A) = |I_m - 2iA\Sigma|^{-\alpha}$ is a characteristic function only if $\alpha \in \{0, \frac{1}{2}, 1,\frac{3}{2},\ldots,(m - 2)/2\} \cup \lbrack(m - 1)/2, \infty)$, we establish a conjecture of Eaton. A similar result is established for the rank 1 noncentral Wishart distribution and is conjecture to also be valid for any greater rank.

Regular variation is used to study the asymptotic behavior of norming operators for generalized domains of attraction. This leads to a powerful decomposition theorem. Applications include a complete, concise description of moment behavior, centering constants, convergence criteria and tail behavior for generalized domains of attraction.

We discuss issues of existence and stochastic modeling in regard to sequences that exhibit combined features of independence and instability of relative frequencies of marginal events. The concept of independence used here is borrowed from the frequentist account of numerical probability advanced by von Mises: A sequence is independent if certain salient asymptotic properties are invariant under the causal selection of subsequences. We show that independence (in the above sense) and instability of relative frequency are indeed compatible and that sequences with such features support stochastic models expressed in terms of envelopes of probability measures.