We are interested in the behavior of Gibbs states at or near the critical temperature. From the point of view of classical probability theory this is a problem about the limit distribution of partial sums of strongly dependent random variables. The problem is very hard in the general case, and almost no rigorous results are known, so we will discuss a special case, Dyson's hierarchical model, in detail. This model can be rigorously investigated, and it may help us to understand the behavior of the more general models. We present the most important results with a sketch of their proofs. Vector-valued models are also discussed, since in this case some new interesting phenomena appear. The last section deals with the translation invariant case. Some recent results are presented and some conjectures and open problems are formulated. The last section can be read independently of the previous sections, but the conjectures formulated there are strongly motivated by the previous results.

Nonequilibrium gradient dynamics of $d$-dimensional particle systems is investigated. The interaction is given by a superstable pair potential of finite range. Solutions are constructed in the well-defined set of locally finite configurations with a logarithmic order of energy fluctuations. If the system is deterministic and $d \leq 2$, then singular potentials are also allowed. For stochastic models with a smooth interaction we need $d \leq 4$. In order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium, we investigate smoothness of the Markov semigroup and describe some properties of its generator.

The infinitely-many-sites model (with no recombination) is reformulated, with sites labelled by elements of [0, 1] and "type" space $E = \lbrack 0, 1\rbrack^{\mathbb{Z}_+}$. A gene is of type $\mathbf{x} = (x_0, x_1,\ldots) \in E$ if $x_0, x_1, \ldots$ is the sequence of sites at which mutations have occurred in the line of descent of that gene. The model is approximated by a diffusion process taking values in $\mathscr{P}^0_a(E)$, the set of purely atomic Borel probability measures $\mu$ on $E$ with the property that the locations of every $n \geq 1$ atoms of $\mu$ form a family tree, and the diffusion is shown to have a unique stationary distribution $\tilde{\mu}$. The principal object of investigation is the $\tilde{\mu}(d\mu)$-expectation of the probability that a random sample from a population with types distributed according to $\mu$ has a given tree structure. Ewens' (1972) sampling formula and Watterson's (1975) segregating-sites distribution are obtained as corollaries.

We prove a strong law of large numbers for the rescaled asymmetric simple exclusion process. By a coupling procedure we show that the density profile is a weak solution to a first order quasilinear partial differential equation. Moreover, the monotonicity of the process allows us to show that it is the unique solution satisfying the entropy condition. The local equilibrium is then an easy consequence.

We consider a Markov additive process $\{(X_n, S_n): n = 0, 1,\ldots\}$, where $\{X_n\}$ is a M.C. on a general state space and $S_n$ is an $\mathbb{R}^d$-valued additive component. Limit theory for $S_n$ is studied via properties of the eigenvalues and eigenfunctions of the kernel of generating functions associated with the transition function of the process. The emphasis is on large deviation theory, but some other limit theorems are also given.

Let $\{(X_n, S_n); n = 0,1,\ldots\}$ be a Markov additive process, $\{X_n\}$ taking values in a general state space $\mathbb{E}$, while $\{S_n\} \subset \mathbb{R}^d$. The large deviation principle is shown to hold for $P_x\{(X_n, S_n) \in A \times n\Gamma\}, A \subset \mathbb{E}, \Gamma \subset \mathbb{R}^d$, the upper bound holding for closed sets $\Gamma$, the lower bound for open sets. The only hypothesis for the lower bound is irreducibility of $\{X_n\}$, and nonsingularity of $\{S_n\}$. The rate function is characterized in terms of the transform kernel of $P_x$.

Let $\mathscr{X}$ be a topological space and $\mathscr{F}$ denote the Borel $\sigma$-field in $\mathscr{X}$. A family of probability measures $\{P_\lambda\}$ is said to obey the large deviation principle (LDP) with rate function $I(\cdot)$ if $P_\lambda(A)$ can be suitably approximated by $\exp\{-\lambda \inf_{x\in A}I(x)\}$ for appropriate sets $A$ in $\mathscr{F}$. Here the LDP is studied for probability measures induced by stochastic processes with stationary and independent increments which have no Gaussian component. It is assumed that the moment generating function of the increments exists and thus the sample paths of such stochastic processes lie in the space of functions of bounded variation. The LDP for such processes is obtained under the weak$^\ast$-topology. This covers a case which was ruled out in the earlier work of Varadhan (1966). As applications, the large deviation principle for the Poisson, Gamma and Dirichlet processes are obtained.

A magnetic body can be considered to consist of $n$ sites, where $n$ is large. The magnetic spins at these $n$ sites, whose sum is the total magnetization present in the body, can be modelled by a triangular array of random variables $(X^{(n)}_1,\ldots, X^{(n)}_n)$. Standard theory of physics would dictate that the joint distribution of the spins can be modelled by $dQ_n(\mathbf{x}) = z^{-1}_n \exp\lbrack -H_n(\mathbf{x})\rbrack\Pi dP(x_j)$, where $\mathbf{x} = (x_1,\ldots, x_n) \in \mathscr{R}^n$, where $H_n$ is the Hamiltonian, $z_n$ is a normalizing constant and $P$ is a probability measure on $\mathscr{R}$. For certain forms of the Hamiltonian $H_n$, Ellis and Newman (1978b) showed that under appropriate conditions on $P$, there exists an integer $r \geq 1$ such that $S_n/n^{1-1/2r}$ converges in distribution to a random variable. This limiting random variable is Gaussian if $r = 1$ and non-Gaussian if $r \geq 2$. In this article, utilizing the large deviation local limit theorems for arbitrary sequences of random variables of Chaganty and Sethuraman (1985), we obtain similar limit theorems for a wider class of Hamiltonians $H_n$, which are functions of moment generating functions of suitable random variables. We also present a number of examples to illustrate our theorems.

We prove a result for small random perturbations of random evolution equations analogous to the Ventsel-Freidlin result on small perturbations of dynamical systems. In particular, we derive large deviations estimates and indicate how they can be used to prove an exit result. The processes we study are governed by equations of the form $dx^\varepsilon(t) = b(x^\varepsilon(t), y(t)) dt + \sqrt \varepsilon \sigma(x^\varepsilon(t)) dw(t)$, where $x^0$ is already a random process. The results include the case where $y$ is an $n$-state Markov process. In the special case $\sigma \equiv Id$, the proof of the estimates is a consequence of a generalization of the "contraction principle" for large deviations: We give sufficient conditions on a continuous function $F$, which ensure that if $\{X_\varepsilon: \varepsilon > 0\}$ satisfies a large deviations principle, then so does $\{F(X_\varepsilon, Y): \varepsilon > 0\}$, where $Y$ is independent of $\{X_\varepsilon: \varepsilon > 0\}$.

We describe simple conditions on the transition density functions of two independent Markov processes $X$ and $Y$ which guarantee the existence of a continuous version for the intersection local time, formally given by $\alpha (z, H) = \int_H\int \delta_z (Y_t - X_s) ds dt$. In the analogous case of self-intersections $\alpha$ can be discontinuous at $z = 0$. We develop a Tanaka-like formula for $\alpha$ and use this to show that the singular part of $\alpha (z,\lbrack 0, T\rbrack^2)$ as $z \rightarrow 0$ is given by $2\int^T_0 U(X_t - z, X_t) dt, a.s.$, where $U$ is the 1-potential of $X$.

Integral-type criteria for thinness and minimal thinness are given for a class of sets. A Kolmogorov-type criterion for Brownian excursions is obtained as a corollary.

For a second-order (hypoelliptic) operator $\mathscr{A} = A_0 + \frac{1}{2} \sum^m_{i=1} A_i$ on a $d$-dimensional manifold $M^d$, let $x_t$ be the diffusion governed by $\mathscr{A}$ and $\varphi (t)$ its associated deterministic control system. We investigate the relations between transience, recurrence and (finite) invariant measures for $x_t$ using the control theoretic decomposition of $M^d$ with respect to $\varphi (t)$. On the invariant control sets for $\varphi (t)$ we obtain the same classification for $x_t$ as is well known for the nondegenerate case, while outside these sets the diffusion $x_t$ is transient.

A new metric is introduced which is suitable for estimating the rate of convergence of processes related to stable random variables. It is shown that it has an upper bound depending on the difference pseudomoments, but not on the absolute moments. This new metric is then applied to get some rates of convergence to a self-similar process constructed from a stable process.

Let $\{Z_\lambda; \lambda = 0, \pm 1,\ldots\}$ be i.i.d. random variables which have a density $f$ which satisfies $f(z) \sim Kz^\alpha\exp\{-z^p\}$ as $z \rightarrow \infty$ for some constants $p > 1, K > 0$, and $\alpha$. Further let $q$ be defined by $p^{-1} + q^{-1} = 1$, and let $\{c_\lambda\}$ be constants with $c_\lambda = O(|\lambda|^{-\theta})$ for some $\theta > \max\{1,2/q\}$. Then, e.g., if $f$ is symmetric $\frac{P(\sum c_\lambda Z_\lambda > z + x/z^{p/q})}{P(\sum c_\lambda Z_\lambda > z)} \rightarrow \exp \{-p\|c\|^{-p}_q x\}, \text{as} z \rightarrow \infty$, for $\|c\|_q = (\sum|c_\lambda|^q)^{1/q}$, and similar results are obtained also for nonsymmetric cases, under some mild further smoothness restrictions. In addition, an order bound for $P(\sum c_\lambda Z_\lambda > z)$ itself is obtained, and precise estimates of this quantity are found for the special case of finite sums. In the companion paper [7], the results are crucially used to study extreme values of moving average processes.

Asymptotic expansions for the distributions of sums of independent nonnegative integer random variables in the neighbourhood of the Poisson distribution are derived, together with explicit estimates of the truncation error. Expansions are also derived for the expectations of at most polynomially growing functions of such sums. Applications to the Poisson binomial and Poisson negative binomial approximations are considered. The method used is an adaptation of the Stein-Chen approach.

Let $X_i$ be a stationary moving average with long-range dependence. Suppose $EX_i = 0$ and $EX^{2n}_i < \infty$ for some $n \geq 2$. When the $X_i$ are Gaussian, then the Hermite polynomials play a fundamental role in the study of noncentral limit theorems for functions of $X_i$. When the $X_i$ are not Gaussian, the relevant polynomials are Appell polynomials. They satisfy a multinomial-type expansion that can be used to establish noncentral limit theorems.

Let $X_n, n \in \mathbb{N}$, be i.i.d. with mean 0, variance 1, and $E(|X_1|^r) < \infty$ for some $r > 3$. Let $B$ be a measurable set such that its distances from the $\sigma$ fields $\sigma (X_1,\ldots, X_n)$ are of order $O(n^{-1/2} (\log n)^{-r/2})$. We prove that for such $B$ the conditional probabilities $P(n^{-1/2}\sum^n_{i=1} X_i \leq t\mid B)$ can be approximated by the standard normal distribution $\Phi (t)$ up to the classical nonuniform bound $(1 + |t|^r)^{-1} n^{-1/2}$. An example shows that this is not true any more if the distances of $B$ from $\sigma (X_1,\ldots, X_n)$ are only of order $O(n^{-1/2}(\log n)^{-r/2+\varepsilon})$ for some $\varepsilon > 0$. For the case $r = 3$ one can obtain the corresponding assertion only under a strengthened assumption.

We have undertaken to develop a new type of stochastic model for nondeterministic empirical processes that exhibit paradoxical characteristics of stationarity, bounded variables, and unstable time averages. By the well-known ergodic theorems of probability theory there is no measure that can model such processes. Hence we are motivated to broaden the scope for mathematical stochastic models. The emerging theory of upper and lower probability, a simple generalization of the theory of finitely additive probability, seems to provide a locus for this new modelling methodology. We focus our attention on the problem of the existence and construction of a lower probability $\underline{P}$ on the power set $2^X$ of a countably infinite product $X$ of a finite set of reals $X_0$, that is shift invariant, monotonely continuous along some class $\mathbf{M}$ of sets that includes the cylinder sets $\mathbf{C}$ and such that $\underline{P}(D^\ast) > 0$ where $D^\ast = \{\mathbf{x} = (x_i)_{i\in Z} \in X: (1/n)\sum^{n-1}_{i=0} x_i$ diverges as $(n \rightarrow \infty)\}$. We show that these constraints are incompatible when $\mathbf{M} = 2^X$, but when $\mathbf{M = C}$ we are able to construct such a lower probability. Most of our results extend to the case of a compact marginal space $X_0$.

Suppose $\mu_1,\ldots,\mu_n$ are probability measures on the same measurable space $(\Omega, \mathscr{F})$. Then if all atoms of each $\mu_i$ have mass $\alpha$ or less, there is a measurable partition $A_1,\ldots, A_n$ of $\Omega$ so that $\mu_i(A_i) \geq V_n(\alpha)$ for all $i = 1,\ldots, n$, where $V_n(\cdot)$ is an explicitly given piecewise linear nonincreasing continuous function on [0, 1]. Moreover, the bound $V_n(\alpha)$ is attained for all $n$ and all $\alpha$. Applications are given to $L_1$ spaces, to statistical decision theory, and to the classical nonatomic case.

For a nondecreasing score function having finitely many jump discontinuities, a representation of $M$-estimators with the second-order asymptotic distribution is established, and the result is also extended to one-step versions of $M$-estimators.

In the so-called secretary problem, if an unknown number, $N$, of options arrive at i.i.d. times with a known continuous distribution, then ignorance of how many options there are becomes almost irrelevant: The optimal rule for infinitely many options is shown to be minimax with respect to all possible distributions of $N$, nearly optimal whenever $N$ is likely to be large, and formal Bayes against a noninformative prior. These results hold whatever the loss function.

Let $u$ be a unitary operator acting in $\mathbb{L}_2(\Omega, F, p)$, where $p$ is a probability measure. We prove that the limit $\lim_{n\rightarrow\infty}\sum_{0 < |k| \leq n} u^k f/k$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$ if and only if the limit $\lim_{n\rightarrow\infty} n^{-1}\sum^{n-1}_{k=0}u^kf$ exists almost surely, for every $f \in \mathbb{L}_2(\Omega, F, p)$.