We consider the threshold voter automaton in one dimension with threshold $\tau > n/2$, where $n$ is the number of neighbors and where we start from a product measure with density $\frac{1}{2}$. It has recently been shown that there is a critical value $\theta_c \approx 0.6469076$, so that if $\tau = \theta n$ with $\theta > \theta_c$ and $n$ is large, then most sites never flip, while for $\theta \in (\frac{1}{2}, \theta_c)$ and $n$ large, there is a limiting state consisting mostly of large regions of points of the same type. Using a supercritical branching process, we show that the behavior at $\theta_c$ differs from both the $\theta > \theta_c$ regime and the $\theta < \theta_c$ regime and that, in some sense, there is a discontinuity both from the left and from the right at this critical value.

We consider the contact process with inhomogeneous deterministic death rates. We prove the following: 1. Such models may have discontinuous transitions, in the sense of surviving at the critical point. 2. If the death rates are identically 1, except on a set which is small enough in a proper sense and where the death rates take a fixed value smaller than 1, then the critical point is identical to that of the homogeneous system. Extensions of the results to other $(d + 1)$-dimensional systems with $d$-dimensional deterministic inhomogeneities are also discussed.

We study a class of attractive spin systems. We prove that for these processes the system dies out when the parameters are on the critical surface. We also prove that the supercritical process survives with positive probability in a sufficiently thick space-time "slab."

We continue our study on the stability properties of a population growth model with sexual reproduction on $\mathbf{Z}^d, d \geq 2$. In the author's previous work, it was proved that in the type IV process (the two-dimensional symmetric model on $\mathbf{Z}^2$), the vacant state $\varnothing$ is stable under perturbation of the initial state (the first kind of perturbation), and it is unstable under perturbation of the birth rate (the second kind of perturbation). In this paper we prove that in the type III process on $\mathbf{Z}^2$, the vacant state $\varnothing$ is stable under the second kind of perturbation, and in three or higher-dimensional symmetric models, the vacant state $\varnothing$ is unstable under the first kind of perturbation. These results, combined with the results obtained earlier, provide a fairly complete picture concerning the stability properties of these models.

We consider the (inhomogeneous) percolation process on $\mathbf{Z}^d \times \mathbf{R}$ defined as follows: Along each vertical line $\{x\} \times \mathbf{R}$ we put cuts at times given by a Poisson point process with intensity $\delta(x)$, and between each pair of adjacent vertical lines $\{x\} \times \mathbf{R}$ and $\{y\} \times \mathbf{R}$ we place bridges at times given by a Poisson point process with intensity $\lambda(x, y)$. We say that $(x, t)$ and $(y, s)$ are connected (or in the same cluster) if there is a path from $(x, t)$ to $(y, s)$ made out of uncut segments of vertical lines and bridges. If we consider only oriented percolation, we have the graphical representation of the (inhomogeneous) $d$-dimensional contact process. We consider these percolation and contact processes in a random environment by taking $\delta = \{\delta(x); x \in \mathbf{Z}^d\}$ and $\lambda = \{\lambda(x,y); x,y \in \mathbf{Z}^d, \|x - y\|_2 = 1\}$ to be independent families of independent identically distributed strictly positive random variables; we use $\delta$ and $\lambda$ for representative random variables. We prove extinction (i.e., no percolation) of these percolation and contact processes, for almost every $\delta$ and $\lambda$, if $\delta$ and $\lambda$ satisfy $\mathbf{E}\{(\log(1 + \lambda))^\beta\} < \infty \text{and} \mathbf{E}\bigg\{\bigg(\log\big(1 + \frac{1}{\delta}\big)\bigg)^\beta\bigg\} < \infty$ for some $\beta > 2d^2\bigg(1 + \sqrt{1 + \frac{1}{d}} + \frac{1}{2d}\bigg),$ and if $\mathbf{E}\bigg\{\bigg(\log\big(1 + \frac{\lambda}{\delta}\big)\bigg)^\beta\bigg\}$ is sufficiently small.

We study equilibrium fluctuations for hydrodynamic limits of a nongradient Ginzburg-Landau model. We prove that the limit fluctuations process is governed by a generalized stationary Ornstein-Uhlenbeck process.

We show that the historical Brownian motion may be recovered from ordinary super-Brownian motion when the dimension $d$ of the underlying Brownian motion is greater than 4. We outline a proof showing that this conclusion is false if $d \leq 3$. The state of affairs in the critical dimension $d = 4$ is left unresolved. Some extensions are given for $1 + \beta$ stable branching mechanisms where $\beta \in (0, 1\rbrack$.

In this article we are mainly concerned with proving lower bounds on some uniform approximation schemes for the local times of one-dimensional Brownian motion. Consequently, this leads to many exact limit theorems for Brownian local times. The latter results are Paul Levy's occupation time approximation, the downcrossing theorem and an intrinsic construction.

We show that the logarithm of the probability that the Brownian sheet has a supremum at most $\epsilon$ over $\lbrack 0, 1\rbrack^2$ is of order $\epsilon^{-2}(\log(1/\epsilon))^3$.

Let $\mathbf{X} = (X_t, t \geq 0)$ be a stationary Gaussian process with zero mean, continuous spectral distribution and twice-differentiable correlation function. An explicit representation is given for the number $N_\psi(T)$ of crossings of a $C^1$ curve $\psi$ by $\mathbf{X}$ on the bounded interval $\lbrack 0, T\rbrack$, in a multiple Wiener-Ito integral expansion. This continues work of the author in which the result was given for $\psi \equiv 0$. The representation is applied to prove new central and noncentral limit theorems for numbers of crossings of constant levels, and some consequences for asymptotic variances are given in mixed-spectrum settings.

Let $\omega = (p_x)_{x\in\mathbb{Z}}$ be an i.i.d. collection of (0, 1)-valued random variables. Given $\omega$, let $(X_n)_{n \geq 0}$ be the Markov chain on $\mathbb{Z}$ defined by $X_0 = 0$ and $X_{n + 1} = X_n + 1(\operatorname{resp}. X_n - 1)$ with probability $p_{X_n}(\operatorname{resp}.1 - p_{X_n})$. It is shown that $X_n/n$ satisfies a large deviation principle with a continuous rate function, that is, $\lim_{n\rightarrow\infty}\frac{1}{n}\log P_\omega(X_n = \lfloor\theta_nn\rfloor) = -I(\theta) \omega-\mathrm{a.s.}\text{for} \theta_n\rightarrow\in\lbrack -1, 1\rbrack.$ First, we derive a representation of the rate function $I$ in terms of a variational problem. Second, we solve the latter explicitly in terms of random continued fractions. This leads to a classification and qualitative description of the shape of $I$. In the recurrent case $I$ is nonanalytic at $\theta = 0$. In the transient case $I$ is nonanalytic at $\theta = -\theta_c, 0, \theta_c$ for some $\theta_c \geq 0$, with linear pieces in between.

Let $T_r$ be the first time a sum $S_n$ of nondegenerate i.i.d. random vectors in $\mathbb{R}^d$ leaves the sphere of radius $r$ in some given norm. We characterize, in terms of the distribution of the individual summands, the following probabilistic behavior: $S_{T_r}/\|S_{T_r}\|$ has no subsequential weak limit supported on a closed half-space. In one dimension, this result solves a very general form of the gambler's ruin problem. We also characterize the existence of degenerate limits and obtain analogous results for triangular arrays along any subsequence $r_k \rightarrow \infty$. Finally, we compute the limiting joint distribution of $(\|S_{T_r}\| - r, S_{T_r}/\|S_{T_r}\|)$.

We consider infinite limit points (in probability) for sums and lightly trimmed sums of i.i.d. random variables normalized by a nonstochastic sequence. More specifically, let $X_1, X_2, \ldots$ be independent random variables with common distribution $F$. Let $M^{(r)}_n$ be the $r$th largest among $X_1, \ldots, X_n$; also let $X^{(r)}_n$ be the observation with the $r$th largest absolute value among $X_1, \ldots, X_n$. Set $S_n = \sum^n_1X_i, ^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n$ and $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n (^{(0)}\tilde{S}_n = ^{(0)}\tilde{S}_n = S_n)$. We find simple criteria in terms of $F$ for $^{(r)}S_n/B_n \rightarrow p \pm \infty$ (i.e., $^{(r)}S_n/B_n$ tends to $\infty$ or to $-\infty$ in probability) or $^{(r)}\tilde{S}_n/B_n \rightarrow p \pm \infty$ when $r = 0, 1, \ldots$. Here $B_n \uparrow \infty$ may be given in advance, or its existence may be investigated. In particular, we find a necessary and sufficient condition for $^{(r)}S_n/n \rightarrow p \infty$. Some equivalences for the divergence of $|^{(r)}\tilde{S}_n|/|X^{(r)}_n|$, or of $^{(r)}S_n/(X^-)^{(s)}_n$, where $(X^-)^{(s)}_n$ is the $s$th largest of the negative parts of the $X_i$, and for the convergence $P\{S_n > 0\}\rightarrow 1$, as $n\rightarrow\infty$, are also proven. In some cases we treat divergence along a subsequence as well, and one such result provides an equivalence for a generalized iterated logarithm law due to Pruitt.

We prove limit theorems for weighted $U$-statistics and express the limit by means of multiple stochastic integrals. This is a generalization of the paper by O'Neil and Redner. In that paper the method of moments was applied which does not work in the general case. Hence we had to work out a different method. In particular, in Theorem 4 we describe the limit of a model proposed by O'Neil and Redner. In this model the weight functions cause an intricate cancellation, and the limit can be presented as a sum of multiple stochastic integrals with different multiplicities.

Let $X$ be a semimartingale and $\Theta$ the space of all predictable $X$-integrable processes $\vartheta$ such that $\int\vartheta dX$ is in the space $\mathscr{S}^2$ of semimartingales. We consider the problem of approximating a given random variable $H \in\mathscr{L}^2$ by a stochastic integral $\int^T_0 \vartheta_s dX_s$, with respect to the $\mathscr{L}^2$-norm. If $X$ is special and has the form $X = X_0 + M + \int \alpha d\langle M\rangle$, we construct a solution in feedback form under the assumptions that $\int \alpha^2 d\langle M\rangle$ is deterministic and that $H$ admits a strong F-S decomposition into a constant, a stochastic integral of $X$ and a martingale part orthogonal to $M$. We provide sufficient conditions for the existence of such a decomposition, and we give several applications to quadratic optimization problems arising in financial mathematics.

Consider the product measure $\mu_p$ on $\{0, 1\}^n$, when 0 $(\operatorname{resp}. 1)$ is given weight $1 - p (\operatorname{resp}. p)$. Consider a monotone subset $A$ of $\{0, 1\}^n$. We give a precise quantitative form to the following statement: if $A$ does not depend much on any given coordinate, $d\mu_p(A)/dp$ is large. Thus, in that case, there is a threshold effect and $\mu_p(A)$ jumps from near 0 to near 1 in a small interval.

For any $n > 2$ we construct an exchangeable sequence of positive continuous random variables, $X_1, \ldots, X_n$, for which, among all stopping rules, $\tau$, based on the $X$'s, $\sup_\tau P\{X_{\tau} = X_1 \vee \cdots \vee X_n\}$ is achieved by a rule based only on the relative ranks of the $X$'s.

Let $(\xi_n)_{n \geq 0}$ be a random walk on a countable group $G$. Sufficient and necessary conditions for the existence of a finite set $A \subseteq G$ and a sequence $g_n \in G$ such that for all natural $n$ we have $P(\xi_n \in A\mid\xi_0 = g_n) = 1$ are presented. This provides a complete solution to the problem of behavior of concentration functions on discrete groups.

Let $R$ be a random time in $\mathscr{F}_\infty$, the terminal element of a filtration $\mathscr{F}_t$ satisfying the usual hypotheses. It is shown that if optimal sampling holds at $R$ for all bounded martingales, then $R$ is optional. If $\mathscr{F}_t$ is the natural pseudo-path filtration of a measurable process $X_t$, then $R$ is optional if (and only if) the conditional distribution of $X_{R + .}$ given $\mathscr{F}_R$ is $Z_R$, where $Z_t$ is an optional version of the conditional distribution of $X_{t +.}$ given $\mathscr{F}_t$.

Upper bounds for the distance in variation between an arbitrary probability measure and the standard normal one are established via some integrodifferential functionals including information. The results are illustrated by gamma- and $t$-distributions. Moreover, as a by-product, another proof of the central limit theorem is obtained.

We introduce the notion of a multivariate local (and tail) empirical process indexed by sets and establish a number of functionals laws of the iterated logarithm for such processes. This leads to a unified methodology to study the almost sure behavior of various statistics which are local functionals of the empirical distribution. Such statistics include density estimators and the Bahadur-Kiefer representation.

We study the class $C_\pi$ of probability measures invariant with respect to the shift transformation on $K^\mathbb{Z}$ (where $K$ is a finite set of integers) which satisfies the Chapman-Kolmogorov equation for a given stochastic matrix $\Pi$. We construct a dense subset of measures in $C_\pi$ distinct from the Markov measure. When $\Pi$ is irreducible and aperiodic, these measures are ergodic but not weakly mixing. We show that the set of measures with infinite memory is $G_\delta$ dense in $C_\pi$ and that the Markov measure is the unique measure which maximizes the Kolmogorov-Sinai (K-S) entropy in $C_\pi$. We give examples of ergodic measures in $C_\pi$ with zero entropy.