We obtain lower bounds for large deviations of vector-valued functionals of a Markov chain with general state space. The bounds are expressed in terms of the convergence parameter of certain kernels. An application to empirical measures of Markov chains is given.

Let $S$ be a finite set and consider the space $\Omega$ of all configurations $\omega: Z^d \rightarrow S$. For $j \in Z^d, \theta_j: \Omega \rightarrow \Omega$ denotes the shift by $j$. Let $V_n$ denote the cube $\{i \in Z^d: 0 \leq i_k < n, 1 \leq k \leq d\}$. Let $\mu$ be a stationary Gibbs measure for a stationary summable interaction. Define $\rho_{V_n}$ as the random probability measure on $\Omega$ given by $\rho_{V_n}(\omega) = n^{-d} \sum_{j \in V_n} \delta_{\theta_j\omega}.$ Our principal result is that the sequence of measures $\mu \circ \rho^{-1}_{V_n}, n = 1,2,\cdots$, satisfies the large deviation principle with normalization $n^d$ and rate function the specific relative entropy $h(\cdot; \mu)$. Applying the contraction principle, we obtain a large deviation principle for the distribution of the empirical distributions; a detailed description of the resulting rate function is provided.

Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random vectors with values in $\mathbb{R}^d, \mu = \mathscr{L}(X_1)$ and let $\lambda$ be the convex conjugate of $\log \hat{\mu}$, where $\hat{\mu}$ is the Laplace transform of $\mu$. For every $d \geq 2$, a probability measure $\mu$ and an open set $A$ in $\mathbb{R}^d$ are constructed so that $\lim \inf_{n\rightarrow\infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) > - \Lambda (A),$ where $S_n = X_1 + \cdots + X_n$ and $\Lambda (A) = \inf_{x \in A} \Lambda (x)$. It is also shown that if $\mu$ satisfies certain regularity conditions, then $\lim \sup_{n\rightarrow \infty} \frac{1}{n} \log P\big(\frac{S_n}{n} \in A\big) \leq - \Lambda (A),$ holds for all Borel sets in $\mathbb{R}^d$.

Some fluctuation results are proved for the volume of the Wiener sausage associated with a $d$-dimensional Brownian motion and a compact set of positive capacity. In high dimensions, the limiting distribution is normal, whereas, if $d = 2$, it is that of a renormalized local time of self-intersections of planar Brownian motion. For $d = 2$ or 3, these limit theorems are closely linked with the renormalization results for self-intersections of Brownian paths.

We give sufficient conditions for local continuity of the isonormal process $L$ at some point of its parameter set. Since a Gaussian process defined on a compact parameter space that is a.s. continuous at each point is sample continuous, our result can be applied to the problem of general sample continuity of Gaussian processes. It is shown that our sufficient conditions are strictly weaker than the classical sufficient conditions for sample continuity.

There exists a Greenian domain $D \subset \mathbb{R}^2$ such that for every set $U$ of attainable minimal Martin boundary points which has null harmonic measure, there exist attainable minimal Martin boundary points $u, \nu \not\in U$ which cannot be connected by an $h$-process in $D$ starting from $u$ and converging to $\nu$.

Let $M$ be an $n$-dimensional Riemannian manifold, $m \in M$ and $T$ be the hitting time of an $r$-sphere around $m$ by Brownian motion $X_t$. We have, for any smooth function $g$ on the unit sphere $S$, under normal coordinates, $E^m \lbrack g(X_T/r) \rbrack = Ig + r^2I(\nu_2g) + r^3 I(\nu_3g) + O(r^4)$ and $E^m \lbrack Tg(X_T/r) \rbrack = E^m \lbrack T \rbrack E^m \lbrack g(X_T/r) \rbrack + r^5c \sum_i \partial_i sI(z_i g) + O(r^6)$, where $I$ is the uniform probability distribution on $S, \nu_2$ and $\nu_3$ are smooth functions on $S$ whose expressions involve scalar curvature, Ricci curvature and their derivatives at $m, c$ is a constant and $s$ is the scalar curvature. $\nu_2 = 0$ if and only if either $n = 2$ or $M$ is an Einstein manifold.

Consider a branching Brownian motion for which the instantaneous branching rate of a particle at position $x$ is given by $\beta(x)$. We assume that $\beta$ is an integrable continuous function converging to 0 as $x \rightarrow \pm \infty$. Let $R(t)$ be the position of the rightmost descendant at the time $t$ of a simple particle starting from position 0 at time 0. We show that there exists a constant $\lambda_0 > 0$ such that $R(t) - \sqrt{\lambda_0/2} t$ converges in distribution as $t \rightarrow \infty$ to a location mixture of the extreme value distribution $\exp (e^{-\sqrt{2\lambda_0 x}})$.

We study the lifetime of a conditioned diffusion (or $h$-path) on a bounded $C^\infty$ domain $G$ in $\mathbb{R}^d$. Making use of results of Donsker and Varadhan, we show that the tail of the distribution of the lifetime decays exponentially; in fact, the decay constant is the same as that for the exponential decay of the tail of the distribution of the first time the unconditioned diffusion exits $G$. In the case of Brownian motion and bounded domains (not necessarily $C^\infty$) we describe some sufficient conditions to ensure the previously described asymptotic results hold here too.

We study the homogenization problem for a diffusion process in a divergence-free random drift field. In particular, in case of a small Gaussian field we derive an asymptotic expansion of the effective diffusion matrix in terms of its spectral density.

In order to analyze the asymptotic behavior of a particle diffusing in a drift field derived from a smooth bounded potential, we develop Nash-type a priori estimates on the transition density of the process. As an immediate consequence of the estimates, we find that for a rapidly decaying potential in $\mathbb{R}^d$, the mean squared displacement behaves like $td + C(t)$, where $\dot{C}(t)$ (the time integral of the "velocity autocorrelation function") decays like $t^{-d/2}$. We also prove, using the estimates, that for a potential in $\mathbb{R}^d$ of the form $V + B$, where $V$ is stationary random ergodic and $B$ has compact support, the diffusion converges under space and time scaling to the same Brownian motion as does the diffusion with $B = 0$.

We consider dependent site percolation on the two-dimensional square lattice, the underlying probability measure being invariant and ergodic under each of the translations and invariant under axis reflections. If this measure satisfies the FKG condition and if percolation occurs, then we show that the infinite occupied cluster is unique with probability 1, and that all vacant star-clusters are finite.

In this paper we show that the phase transition in the contact process manifests itself in the behavior of large finite systems. To be precise, if we let $\sigma_N$ denote the time the process on $\{1, \cdots, N\}$ first hits $\varnothing$ starting from all sites occupied, then there is a critical value $\lambda_c$ so that (i) for $\lambda < \lambda_c$ there is a constant $\gamma(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty, \sigma_n /\log N \rightarrow 1/\gamma(\lambda)$ in probability and (ii) for $\lambda > \lambda_c$ there are constants $\alpha (\lambda), \beta(\lambda) \in (0, \infty)$ so that as $N \rightarrow \infty$, $P(\alpha(\lambda)/2 - \varepsilon \leq (\log \sigma_N)/N \leq \beta (\lambda) + \varepsilon) \rightarrow 1,$ for all $\varepsilon > 0$. Our results improve upon an earlier work of Griffeath but as the reader can see the second one still needs improvement. To help decide what should be true for the contact process we also consider the analogous problem for the biased voter model. For this process we can show $(\log \sigma_N)/N \rightarrow \alpha(\lambda) = \beta(\lambda)$ in probability, and it seems likely that the same result is true for the contact process.

We improve the complete convergence theorem for the contact process in high dimensions by enlarging the range of the infection parameter in the hypothesis of that theorem.

Let $\{X_j: j \in \mathbb{N}\}$ be a sequence of associated random variables with zero mean and let $r > 2$. We give two conditions--on the moments and on the covariance structure of the process--which guarantee that $\sup_{m \in \mathbb{N} \cup \{0\}} E| \sum^{m+n}_{j=m+1} X_j|^r = O(n^{r/2})$ holds. Examples show that neither condition can be weakened.

We prove a conjecture of Joag-Dev and Goel that if $M = M(\sigma) = \{i: \sigma(i) = i\}$ is the (random) match set, or set of fixed points, of a random permutation $\sigma$ of $1,2,\ldots, n$, then $f(M)$ and $g(M)$ are positively correlated whenever $f$ and $g$ are increasing real-valued set functions on $2^{\{1,\ldots, n\}}$, i.e., $Ef(M)g(M) \geq Ef(M)Eg(M)$. No simple use of the FKG or Ahlswede-Daykin inequality seems to establish this, despite the fact that the FKG hypothesis is "almost" satisfied. Instead we reduce to the case where $f$ and $g$ take values in $\{0, 1\}$, and make a case-by-case argument: Depending on the specific form of $f$ and $g$, we move the probability weights around so as to make them satisfy the FKG or Ahlswede-Daykin hypotheses, without disturbing the expectations $Ef, Eg, Efg$. This approach extends the methodology by which FKG-style correlation inequalities can be proved.

Upper and lower bounds are given on the moment generating function of the time taken by a Markov chain to visit at least $n$ of $N$ selected subsets of its state space. An example considered is the class of random walks on the symmetric group that are constant on conjugacy classes. Application of the bounds yields, for example, the asymptotic distribution of the time taken to see all $N!$ arrangements of $N$ cards as $N\rightarrow\infty$ for certain shuffling schemes.

A random walk on an infinite tree is given a particular kind of positive feedback so edges already traversed are more likely to be traversed in the future. Using exchangeability theory, the process is shown to be equivalent to a random walk in a random environment (RWRE), that is to say, a mixture of Markov chains. Criteria are given to determine whether a RWRE is transient or recurrent. These criteria apply to show that the reinforced random walk can vary from transient to recurrent, depending on the value of an adjustable parameter measuring the strength of the feedback. The value of the parameter at the phase transition is calculated.

Using a Gaussian randomization technique, we prove that a random variable $X$ with values in a Banach space $B$ satisfies the (compact) law of the iterated logarithm if and only if (i) $E(\|X\|^2/LL\|X\|) < \infty$, (ii) $\{|\langle x^\ast, X \rangle |^2; x^\ast \in B^\ast, \|x^\ast\| \leq 1\}$ is uniformly integrable and (iii) $S_n(x)/a_n\rightarrow 0$ in probability. In particular, if $B$ is of type 2, in order that $X$ satisfy the law of the iterated logarithm it is necessary and sufficient that $X$ have mean zero and satisfy (i) and (ii). The proof uses tools of the theory of Gaussian random vectors as well as by now classical arguments of probability in Banach spaces. It also sheds some light on the usual law of the iterated logarithm on the line.

It is shown that "studentizing," i.e., normalizing by the sample standard deviation rather than the population standard deviation, can improve the rate of convergence in the central limit theorem. This provides concise confirmation of one feature of the folklore that a studentized sum is in some sense more robust than a normed sum. The case of infinite population standard deviation is also examined.

A "law of large numbers" for the maximum of i.i.d. univariate normal random variables is extended to a general multivariate case. Let $\mathbf{Z}_i$ denote i.i.d. Banach space valued random variables with a centered Gaussian distribution. Let $\mathbf{K}$ denote the unit ball of the reproducing kernel Hilbert space. Then with probability 1, the maximum distance from the sample points $\mathbf{Z}_1, \mathbf{Z}_2,\ldots, \mathbf{Z}_n$ to the set $\sqrt{2 \log n} \mathbf{K}$ approaches zero. In addition, the sample forms epsilon nets for this set as $n$ tends to infinity.

A $U$-process is a collection of $U$-statistics indexed by a family of symmetric kernels. In this paper, two functional limit theorems are obtained for sequences of standardized $U$-processes. In one case the limit process is Gaussian; in the other, the limit process has finite dimensional distributions of infinite weighted sums of $\chi^2$ random variables. Goodness-of-fit statistics provide examples.

In this paper we give a simple construction of the general stationary regenerative set, based on the stationary version of the associated subordinator (increasing Levy process). We show that, in a certain sense, the closed range of such a Levy process is a stationary regenerative subset of $\mathbb{R}$. The distribution of this regenerative set is $\sigma$-finite in general; it is finite $\operatorname{iff}$ the increments of the Levy process have finite expectation.

We consider regenerative systems on the real line and study their structure. Local times and exit systems are defined. This leads to time changes and last exit decompositions. We establish a correspondence between stationary regenerative systems and Markov additive processes which are stationary in the space component.

Let $\mathbf{\alpha}_n$ be a sequence of i.i.d. nondecreasing random maps on a subset $S$ of $\mathbb{R}^k$ into itself and let $X_0$ be a random variable with values in $S$ independent of the sequence $\mathbf{\alpha}_n$. Then $X_n \equiv \mathbf{\alpha}_n \cdots \mathbf{\alpha}_1X_0$ is a Markov process. Conditions for the existence of unique invariant probabilities are obtained for such Markov processes which are not in general irreducible, extending earlier results of Dubins and Freedman to multidimensional and noncompact state spaces. In addition, a functional central limit theorem is obtained. These yield new results in time series and economic models.

For the derivatives $f^{(k)}_\alpha(x)$ of the one-sided stable density of index $\alpha \in (0, 1)$ asymptotic formulas are computed as $k \rightarrow \infty$ thereby exhibiting the detailed analytic structure for large orders of derivatives. The results extend those for the well-known case $\alpha = \frac{1}{2}$ which may be expressed in terms of Laguerre polynomials (formulas of Plancherel-Rotach type).

Let $(T, \Theta)$ be a compact measurable topological space and $\Psi_q(x) = \exp|x|^q - 1, 1 \leq q < \infty$. Let $X = \{X(\omega, t), \omega \in \Omega, t \in T\}$ be a $\Theta$-measurable stochastic process such that $\|X(s) - X(t)\|_{L^\Psi q(\Omega)}\leq d(s, t)$ for every $(s, t) \in T \otimes T$, where $d(\cdot, \cdot)$ is some continuous pseudometric on $(T, \Theta)$. We give a sufficient condition expressed in terms of a majorizing measure on $(T, d)$ in order that $X$ take values in the Orlicz space $L^{\Psi_q}(T, \mu)$, where $q \leq q' < \infty$ and $\mu$ any Borel probability measure on $(T, \Theta)$.