The subject of this article is a class of measure-valued Markov processes. A typical example is super-Brownian motion. The Laplacian $\Delta$ plays a fundamental role in the theory of Brownian motion. For super-Brownian motion, an analogous role is played by the operator $\Delta u - \psi(u)$, where a nonlinear function $\psi$ describes the branching mechanism. The class of admissible functions $\psi$ includes the family $\psi(u) = u^\alpha, 1 < \alpha \leq 2$. Super-Brownian motion belongs to the class of continuous state branching processes investigated in 1968 in a pioneering work of Watanabe. Path properties of super-Brownian motion are well known due to the work of Dawson, Perkins, Le Gall and others. Partial differential equations involving the operator $\Delta u - \psi(u)$ have been studied independently by several analysts, including Loewner and Nirenberg, Friedman, Brezis, Veron, Baras and Pierre. Connections between the probabilistic and analytic theories have been established recently by the author.

It is shown that a regular stationary random field on $\mathbf{Z}^2$ exhibits the weak (strong) commutation property if and only if its spectral density is the squared modulus of a weakly (strongly) outer function in the Hardy space $H^2(\mathbf{T}^2)$ of the torus. Applications to prediction are discussed.

In this paper, the authors show that the smallest (if $p \leq n$) or the $(p - n + 1)$-th smallest (if $p > n$) eigenvalue of a sample covariance matrix of the form $(1/n)XX'$ tends almost surely to the limit $(1 - \sqrt y)^2$ as $n \rightarrow \infty$ and $p/n \rightarrow y \in (0,\infty)$, where $X$ is a $p \times n$ matrix with iid entries with mean zero, variance 1 and fourth moment finite. Also, as a by-product, it is shown that the almost sure limit of the largest eigenvalue is $(1 + \sqrt y)^2$, a known result obtained by Yin, Bai and Krishnaiah. The present approach gives a unified treatment for both the extreme eigenvalues of large sample covariance matrices.

Let $\Lambda$ be the set of probability measures $\lambda$ on $\lbrack 0,1\rbrack$. Let $M_n = \{(c_1,\ldots,c_n)\mid\lambda \in \Lambda\}$, where $c_k = c_k(\lambda) = \int^1_0x^k d\lambda, k = 1,2,\ldots$ are the ordinary moments, and assign to the moment space $M_n$ the uniform probability measure $P_n$. We show that, as $n \rightarrow \infty$, the fixed section $(c_1,\ldots,c_k)$, properly normalized, is asymptotically normally distributed. That is, $\sqrt n\lbrack(c_1,\ldots,c_k) - (c^0_1,\ldots,c^0_k)\rbrack$ converges to $\mathrm{MVN}(0,\Sigma)$, where $c^0_i$ correspond to the arc sine law $\lambda_0$ on $\lbrack 0,1\rbrack$. Properties of the $k \times k$ matrix $\Sigma$ are given as well as some further discussion.

It is shown that the rate of convergence in the von Mises conditions of extreme value theory determines the distance of the underlying distribution function $F$ from a generalized Pareto distribution. The distance is measured in terms of the pertaining densities with the limit being ultimately attained if and only if $F$ is ultimately a generalized Pareto distribution. Consequently, the rate of convergence of the extremes in an iid sample, whether in terms of the distribution of the largest order statistics or of corresponding empirical truncated point processes, is determined by the rate of convergence in the von Mises condition. We prove that the converse is also true.

Let $G$ be a finite graph or an infinite graph on which $\mathbb{Z}^d$ acts with finite fundamental domain. If $G$ is finite, let $\mathbf{T}$ be a random spanning tree chosen uniformly from all spanning trees of $G$; if $G$ is infinite, methods from Pemantle show that this still makes sense, producing a random essential spanning forest of $G$. A method for calculating local characteristics (i.e., finite-dimensional marginals) of $\mathbf{T}$ from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix ($n$-dimensional minors) are needed to compute small ($n$-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for $G$, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when $G$ is an infinite graph admitting a $Z^d$-action with finite quotient. The same computation also gives the entropy of the law of $\mathbf{T}$. These results are applied to the problem of tiling certain lattices by dominos--the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. Included is a generalization of moments to tree-valued random variables and criteria for these generalized moments to determine a distribution.

We establish hydrodynamic limits for a class of attractive, reversible particle systems with an infinite range of interaction. The limiting nonlinear diffusion equations have diffusion coefficients which are functions of the local density, and which have a singularity at a critical value of the density. On open driven systems, these singular diffusion limits explain the observed nontrivial scaling behavior known as self-organized criticality.

For continuum $1/r^2$ Ising models, we prove that the critical value of the long range coupling constant (inverse temperature), above which an ordered phase occurs (for strong short range cutoff), is exactly 1. This leads to a proof of the existence of an ordered phase with slow decay of correlations. Our arguments involve comparisons between continuum and discrete Ising models, including (quenched and annealed) site diluted models, which may be of independent interest.

We prove laws of large numbers for a cellular automaton in the space $\{0,1,\ldots,p - 1\}^Z$ with $p$ being a prime number. The dynamics $\tau$ of the system are defined by $\tau\eta(x) = \eta(x - 1) + \eta(x + 1) \operatorname{mod} p$ for $\eta \in X$.

Assign to each edge $e$ of the square lattice $\mathbb{Z}^2$ a random bond conductivity $c(e)$. If $c(e)$ are stationary, ergodic and such that $0 < a < c(e) < b < \infty$ for all edges $e$, then there is a central limit theorem for the corresponding reversible random walk on the lattice which holds for almost all environments.

In a set of even cardinality $n$, each member ranks all the others in order of preference. A stable matching is a partition of the set into $n/2$ pairs, with the property that no two unpaired members both prefer each other to their partners under matching. It is known that for some problem instances no stable matching exists. What if an instance of the ranking system is chosen uniformly at random? We show that the mean and the variance of the total number of stable matchings for the random problem instance are asymptotic to $e^{1/2}$ and $(\pi n/4e)^{1/2}$, respectively. Consequently, $\operatorname{Prob}$ a stable matching exists) $\gtrsim (4e^3/\pi n)^{1/2}$. We also prove that, given the last event, in every stable matching the sum of the ranks of all members (as rank ordered by their partners) is asymptotic to $n^{3/2}$, and the largest rank of a partner is of order $n^{1/2}\log n$, with superpolynomially high conditional probability. In other words, stable partners are very likely to be relatively close to the tops of each other's preference lists.

A continuous version of De Finetti's theorem is proved in which the role of the homogeneous product states is played by the independent increment stationary processes on the real line. The proof is based on a conditional, finite De Finetti's theorem (i.e., a result involving only a finite number of random variables and exchangeable conditional expectations rather than exchangeable probabilities). Our technique of proof improves and simplifies a result of Freedman and includes a generalization of the quantum De Finetti's theorem as well as some more recent variants of it. The last section of the paper is an attempt to answer a question of Diaconis and Freedman.

Necessary and sufficient conditions for the law of large numbers and sufficient conditions for the central limit theorem for $U$-processes are given. These conditions are in terms of random metric entropies. The CLT and LLN for VC subgraph classes of functions as well as for classes satisfying bracketing conditions follow as consequences of the general results. In particular, Liu's simplicial depth process satisfies both the LLN and the CLT. Among the techniques used, randomization, decoupling inequalities, integrability of Gaussian and Rademacher chaos and exponential inequalities for $U$-statistics should be mentioned.

For any positive integer $n$, there exists a unimodal distribution $\mu$ such that $\mu \ast \mu$ is $n$-modal. Furthermore, there is a unimodal distribution $\mu$ such that $\mu \ast \mu$ has infinitely many modes. Lattice analogues of the results are also given.

A representation of one-dependent processes is given in terms of Hilbert spaces, vectors and bounded linear operators on Hilbert spaces. This generalizes a construction of one-dependent processes that are not two-block-factors. We show that all one-dependent processes admit a representation. We prove that if there is in the Hilbert space a closed convex cone that is invariant under certain operators and that is spanned by a finite number of linearly independent vectors, then the corresponding process is a two-block-factor of an independent process. Apparently the difference between two-block-factors and non-two-block-factors is determined by the geometry of invariant cones. The dimension of the smallest Hilbert space that represents a process is a measure for the complexity of the structure of the process. For two-valued one-dependent processes, if there is a cylinder with measure equal to zero, then this process can be represented by a Hilbert space with dimension smaller than or equal to the length of this cylinder. In the two-valued case a cylinder (with measure equal to zero) whose length is minimal and less than or equal to 7 is symmetric. We generalize the concept of Hilbert space representation to $m$-dependent processes and it turns out that all $m$-dependent processes admit a representation. Several theorems can be generalized to $m$-dependent processes.

Let $S$ be a compact metric space, let $\theta \geq 0$, and let $\nu_0$ be a Borel probability measure on $S$. An explicit formula is found for the transition function of the Fleming-Viot process with type space $S$ and mutation operator $(Af)(x) = (1/2)\theta\int_S(f(\xi) - f(x))\nu_0(d\xi)$.

Let $X$ be a Markov process on an interval $E$ of $\mathbb{R}$, with lifetime $\zeta$, admitting a local time at each point and such that $P_x(X$ hits $y) > 0$ for all $x,y$ in $E$. We prove here that the local times process $(L^x_\zeta, x \in E)$ is a Markov process if and only if $X$ has fixed birth and death points and $X$ has continuous paths. The sufficiency of this condition has been established by Ray, Knight and Walsh. The necessity is proved using arguments based on excursion theory. This result has been proved before in Eisenbaum and Kaspi for symmetric processes using the existence of a zero mean Gaussian process with the Green function as covariance.

In this work we study local times for a class of measure-valued Markov processes known as superprocesses. We begin by deriving analogues of well-known properties of ordinary local times. Then, restricting our attention to a class of superprocesses (which includes the important case of super-Brownian motion), we prove more detailed properties of the local times, such as joint continuity and a global Holder condition. These are then used to obtain path properties of the superprocesses themselves. For example, we compute the Hausdorff dimension of the "level sets" of super-Brownian motion.

In a generalized two-color Polya urn scheme, allowing negative replacements, we use martingale techniques to obtain weak invariance principles for the urn process $(W_n)$, where $W_n$ is the number of white balls in the urn at stage $n$. The normalizing constants and the limiting Gaussian process are shown to depend on the ratio of the eigenvalues of the replacement matrix.

Motivated by recent results on pathwise central limit theorems, we study in a systematic way log-average versions of classical limit theorems. For partial sums $S_k$ of independent r.v.'s we prove under mild technical conditions that $(1/\log N)\sum_{k \leq N}(1/k)I\{S_k/a_k \in \cdot\} \rightarrow G(\cdot)$ (a.s.) if and only if $(1/\log N)\sum_{k \leq N}(1/k)P(S_k/a_k \in \cdot) \rightarrow G(\cdot)$. A functional version of this result also holds. For partial sums of i.i.d. r.v.'s attracted to a stable law, we obtain a pathwise version of the stable limit theorem as well as a strong approximation by a stable process on log dense sets of integers. We also give necessary and sufficient conditions for the law of large numbers in log density.

Most large deviation results give asymptotic expressions for $\log P(Y_n \geq y_n)$, where the event $\{Y_n \geq y_n\}$ is a large deviation event, that is, $P(Y_n \geq y_n)$ goes to 0 exponentially fast. We refer to such results as weak large deviation results. In this paper we obtain strong large deviation results for arbitrary random variables $\{Y_n\}$, that is, we obtain asymptotic expressions for $P(Y_n \geq y_n)$, where $\{Y_n \geq y_n\}$ is a large deviation event. These strong large deviation results are obtained for lattice valued and nonlattice valued random variables and require some conditions on their moment generating functions. These results strengthen existing results which apply mainly to sums of independent and identically distributed random variables. Since $Y_n$ may not possess a probability density function, we consider the function $q_n(y; b_n,S) = \lbrack(b_n/\mu(S))P(b_n(Y_n - y) \in S)\rbrack$, where $b_n \rightarrow \infty, \mu$ is the Lebesgue measure on $R$, and $S$ is a measurable subset of $R$ such that $0 < \mu(S) < \infty$. The function $q_n(y; b_n,S)$ is the p.d.f. of $Y_n + Z_n$, where $Z_n$ is uniform on $-S/b_n$, and will be called the pseudodensity function of $Y_n$. By a local limit theorem we mean the convergence of $q_n(y_n; b_n,S)$ as $n \rightarrow \infty$ and $y_n \rightarrow y^\ast$. In this paper we obtain local limit theorems for arbitrary random variables based on easily verifiable conditions on their characteristic functions. These local limit theorems play a major role in the proofs of the strong large deviation results of this paper. We illustrate these results with two typical applications.

Let $\alpha_n$ be the classical empirical process. Assume $T$, defined on $D\lbrack 0, 1\rbrack$, satisfies the Lipschitz condition with respect to a weighted sup-norm in $D\lbrack 0, 1\rbrack$. Explicit bounds for $P(T(\alpha_n) \geq x_n\sqrt n)$ are obtained for every $n \geq n_0$ and all $x_n \in (0, \sigma\rbrack$, where $n_0$ and $\sigma$ are also explicitly given. These bounds lead to moderately large deviations and expansions of the asymptotic large deviations for $T(\alpha_n)$. The present theory closely relates large and moderately large deviations to tails of the asymptotic distributions of considered statistics. It unifies and generalizes some earlier results. In particular, some results of Groeneboom and Shorack are easily derived.

Let $(X_i)_{i\in\mathbb{Z}^d_+}$ be an array of zero-mean independent identically distributed random vectors with values in $\mathbb{R}^k$ with finite variance, and let $\mathscr{L}$ be a class of Borel subsets of $\lbrack 0, 1\rbrack^d$. If, for the usual metric, $\mathscr{L}$ is totally bounded and has a convergent entropy integral, we obtain a strong invariance principle for an appropriately smoothed version of the partial-sum process $\{\sum_{i\in\nu S} X_i: S \in \mathscr{L}\}$ with an error term depending only on $\mathscr{L}$ and on the tail distribution of $X_1$. In particular, when $\mathscr{L}$ is the class of subsets of $\lbrack 0, 1\rbrack^d$ with $\alpha$-differentiable boundaries introduced by Dudley, we prove that our result is optimal.

The nonlinear filtering problem is studied for models where the samples of the signal and the noise are elements of some general abstract Wiener space. The signal is allowed to depend on the noise and the optimal filter is expressed as an explicit functional of the observed sample (trajectory). It is shown that this functional satisfies the Zakai equation. As a necessary technical tool, a class of shift transformations on the Wiener space is studied and an analog of Cameron-Martin-Girsanov's theorem is obtained.