We consider a system of interacting Brownian particles in ℝ^d with a pairwise potential, which is radially symmetric, of finite range and attains a unique minimum when the distance of two particles becomes a>0. The asymptotic behavior of the system is studied under the zero temperature limit from both microscopic and macroscopic aspects. If the system is rigidly crystallized, namely if the particles are rigidly arranged in an equal distance a, the crystallization is kept under the evolution in macroscopic time scale. Then, assuming that the crystal has a definite limit shape under a macroscopic spatial scaling, the translational and rotational motions of such shape are characterized.

We study the zero temperature limit for interacting Brownian particles in one dimension with a pairwise potential which is of finite range and attains a unique minimum when the distance of two particles becomes a>0. We say a chain is formed when the particles are arranged in an “almost equal” distance a. If a chain is formed at time 0, so is for positive time as the temperature of the system decreases to 0 and, under a suitable macroscopic space-time scaling, the center of mass of the chain performs the Brownian motion with the speed inversely proportional to the total mass. If there are two chains, they independently move until the time when they meet. Then, they immediately coalesce and continue the evolution as a single chain. This can be extended for finitely many chains.

We are interested in stationary “fluid” random evolutions with independent increments. Under some mild assumptions, we show they are solutions of a stochastic differential equation (SDE). There are situations where these evolutions are not described by flows of diffeomorphisms, but by coalescing flows or by flows of probability kernels.

In an intermediate phase, for which there exist a coalescing flow and a flow of kernels solution of the SDE, a classification is given: All solutions of the SDE can be obtained by filtering a coalescing motion with respect to a subnoise containing the Gaussian part of its noise. Thus, the coalescing motion cannot be described by a white noise.

Precise asymptotics for moderate deviation probabilities are established for open convex sets in both the finite- and infinite-dimensional settings. Our results are based on the existence of dominating points for these sets, a related representation formula, and asymptotics for the integral term in this formula.

We prove that the convolution of a selfdecomposable distribution with its background driving law is again selfdecomposable if and only if the background driving law is s-selfdecomposable. We will refer to this as the factorization property of a selfdecomposable distribution; let L^f denote the set of all these distributions. The algebraic structure and various characterizations of L^f are studied. Some examples are discussed, the most interesting one being given by the Lévy stochastic area integral. A nested family of subclasses L^f_n, n≥0, (or a filtration) of the class L^f is given.

We present two path decompositions of Markov chains (with general state space) by means of harmonic functions, which are dual to each other. They can be seen as a generalization of Williams’ decomposition of a Brownian motion with drift. The results may be illustrated by a multitude of examples, but we confine ourselves to different types of random walks and the Pólya urn.

We prove a uniform functional law of the logarithm for the local empirical process. To accomplish this we combine techniques from classical and abstract empirical process theory, Gaussian distributional approximation and probability on Banach spaces. The body of techniques we develop should prove useful to the study of the strong consistency of d-variate kernel-type nonparametric function estimators.

The leading term in the normal approximation to the distribution of Student’s t statistic is derived in a general setting, with the sole assumption being that the sampled distribution is in the domain of attraction of a normal law. The form of the leading term is shown to have its origin in the way in which extreme data influence properties of the Studentized sum. The leading-term approximation is used to give the exact rate of convergence in the central limit theorem up to order n^−1/2, where n denotes sample size. It is proved that the exact rate uniformly on the whole real line is identical to the exact rate on sets of just three points. Moreover, the exact rate is identical to that for the non-Studentized sum when the latter is normalized for scale using a truncated form of variance, but when the corresponding truncated centering constant is omitted. Examples of characterizations of convergence rates are also given. It is shown that, in some instances, their validity uniformly on the whole real line is equivalent to their validity on just two symmetric points.

We study the partial maxima of stationary α-stable processes. We relate their asymptotic behavior to the ergodic theoretical properties of the flow. We observe a sharp change in the asymptotic behavior of the sequence of partial maxima as flow changes from being dissipative to being conservative, and argue that this may indicate a change from a short memory process to a long memory process.

The Lauricella theory of multiple hypergeometric functions is used to shed some light on certain distributional properties of the mean of a Dirichlet process. This approach leads to several results, which are illustrated here. Among these are a new and more direct procedure for determining the exact form of the distribution of the mean, a correspondence between the distribution of the mean and the parameter of a Dirichlet process, a characterization of the family of Cauchy distributions as the set of the fixed points of this correspondence, and an extension of the Markov–Krein identity. Moreover, an expression of the characteristic function of the mean of a Dirichlet process is obtained by resorting to an integral representation of a confluent form of the fourth Lauricella function. This expression is then employed to prove that the distribution of the mean of a Dirichlet process is symmetric if and only if the parameter of the process is symmetric, and to provide a new expression of the moment generating function of the variance of a Dirichlet process.

Concentration of measure is studied, and obtained, for stable and related random vectors.

Let (L,≼) be a finite distributive lattice, and suppose that the functions f₁,f₂:L→ℝ are monotone increasing with respect to the partial order ≼. Given μ a probability measure on L, denote by $\mathbb{E}(f_{i})$ the average of f_i over L with respect to μ, i=1,2. Then the FKG inequality provides a condition on the measure μ under which the covariance, $\operatorname{Cov}(f_{1},f_{2}):=\mathbb{E}(f_{1}f_{2})-\mathbb{E}(f_{1})\mathbb{E}(f_{2})$ , is nonnegative. In this paper we derive a “third-order” generalization of the FKG inequality: Let f₁, f₂ and f₃ be nonnegative, monotone increasing functions on L; and let μ be a probability measure satisfying the same hypotheses as in the classical FKG inequality; then $$\begin{array}{l}2\mathbb{E}(f_{1}f_{2}f_{3})\\\qquad{}-[\mathbb{E}(f_{1}f_{2})\mathbb{E}(f_{3})+\mathbb{E}(f_{1}f_{3})\mathbb{E}(f_{2})+\mathbb{E}(f_{1})\mathbb{E}(f_{2}f_{3})]\\\qquad{}+\mathbb{E}(f_{1})\mathbb{E}(f_{2})\mathbb{E}(f_{3})\end{array}$$ is nonnegative. This result reduces to the FKG inequality for the case in which f₃≡1.

We also establish fourth- and fifth-order generalizations of the FKG inequality and formulate a conjecture for a general mth-order generalization. For functions and measures on ℝⁿ we establish these inequalities by extending the method of diffusion processes. We provide several applications of the third-order inequality, generalizing earlier applications of the FKG inequality. Finally, we remark on some connections between the theory of total positivity and the existence of inequalities of FKG-type within the context of Riemannian manifolds.

Using the Wiener–Hopf factorization, it is shown that it is possible to bound the path of an arbitrary Lévy process above and below by the paths of two random walks. These walks have the same step distribution, but different random starting points. In principle, this allows one to deduce Lévy process versions of many known results about the large-time behavior of random walks. This is illustrated by establishing a comprehensive theorem about Lévy processes which converge to ∞ in probability.

We study the Fourier expansion of the distribution density of a Lévy process in a compact Lie group based on the Peter–Weyl theorem.

The sharp asymptotics for the L²-quantization errors of Gaussian measures on a Hilbert space and, in particular, for Gaussian processes is derived. The condition imposed is regular variation of the eigenvalues.

A weighted U-statistic based on a random sample X₁,…,X_n has the form U_n=∑_1≤i,j≤nw_i−jK(X_i,X_j), where K is a fixed symmetric measurable function and the w_i are symmetric weights. A large class of statistics can be expressed as weighted U-statistics or variations thereof. This paper establishes the asymptotic normality of U_n when the sample observations come from a nonlinear time series and linear processes.

A stochastic Navier–Stokes equation with space-time Gaussian white noise is considered, having as infinitesimal invariant measure a Gaussian measure μ_ν whose covariance is given in terms of the enstrophy. Pathwise uniqueness for μ_ν-a.e. initial velocity is proven for solutions having μ_ν as invariant measure.

In a celebrated work by Hoeffding [ J. Amer. Statist. Assoc. 58 (1963) 13–30], several inequalities for tail probabilities of sums M_n={X}₁+⋯+{X}_n of bounded independent random variables X_j were proved. These inequalities had a considerable impact on the development of probability and statistics, and remained unimproved until 1995 when Talagrand [Inst. Hautes Études Sci. Publ. Math. 81 (1995a) 73–205] inserted certain missing factors in the bounds of two theorems. By similar factors, a third theorem was refined by Pinelis [Progress in Probability 43 (1998) 257–314] and refined (and extended) by me. In this article, I introduce a new type of inequality. Namely, I show that ℙ{M_n≥x}≤cℙ{S_n≥x}, where c is an absolute constant and S_n={ɛ}₁+⋯+{ɛ}_n is a sum of independent identically distributed Bernoulli random variables (a random variable is called Bernoulli if it assumes at most two values). The inequality holds for those x∈ℝ where the survival function x↦ℙ{S_n≥x} has a jump down. For the remaining x the inequality still holds provided that the function between the adjacent jump points is interpolated linearly or log-linearly. If it is necessary, to estimate ℙ{S_n≥x} special bounds can be used for binomial probabilities. The results extend to martingales with bounded differences. It is apparent that Theorem 1.1 of this article is the most important. The inequalities have applications to measure concentration, leading to results of the type where, up to an absolute constant, the measure concentration is dominated by the concentration in a simplest appropriate model, such results will be considered elsewhere.

Approximations to sums of stationary and ergodic sequences by martingales are investigated. Necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained. It is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a Lindeberg–Feller condition. Using the explicit construction of the approximating martingales, a central limit theorem is derived for the sample means of linear processes. The conditions are not sufficient for the functional version of the central limit theorem. This is shown by an example, and a slightly stronger sufficient condition is given.

We study the relative entropy density for generalized Gibbs measures. We first show its existence and obtain a familiar expression in terms of entropy and relative energy for a class of “almost Gibbsian measures” (almost sure continuity of conditional probabilities). For quasilocal measures, we obtain a full variational principle. For the joint measures of the random field Ising model, we show that the weak Gibbs property holds, with an almost surely rapidly decaying translation-invariant potential. For these measures we show that the variational principle fails as soon as the measures lose the almost Gibbs property. These examples suggest that the class of weakly Gibbsian measures is too broad from the perspective of a reasonable thermodynamic formalism.