Let $S$ be the set of $q \times q$ matrices with positive entries, such that each column and each row contains a strictly positive element, and denote by $S^\circ$ the subset of these matrices, all entries of which are strictly positive. Consider a random ergodic sequence $(X_n)_{n \geq1}$ in $S$. The aim of this paper is to describe the asymptotic behavior of the random products $X^{(n)} =X_n \ldots X _1, n\geq 1$ under the main hypothesis $P(\Bigcup_{n\geq 1}[X^{(n)}\in S^\circ])>0$. We first study the behavior “in direction” of row and column vectors of $X^{(n)}$. Then, adding a moment condition, we prove a law of large numbers for the entries and lengths of these vectors and also for the spectral radius of $X^{(n)}$ . Under the mixing hypotheses that are usual in the case of sums of real random variables, we get a central limit theorem for the previous quantities. The variance of the Gaussian limit law is strictly positive except when $(X^{(n)})_{n\geq 1}$ is tight. This tightness property is fully studied when the $X_n, n\geq 1$, are independent.

Let $\{X_i\}_ {i \geq 1}$ be i.i.d. random variables with common distribution function $F$, and let $S_n=\sum_1^n X_i$. We find a necessary and sufficient condition (directly in terms of$F$) for the existence of sequences of constants $\{\alpha_n\}$ and $\{\beta_n\}$ with $\beta_n\uparrow\infty$ such that $0<\liminf \beta_n^{-1}\max_{j\leq n}|S_j- \alpha_j|<\infty$ w.p.1. and such that for any choice of $\tilde{\alpha}_n$, it holds w.p.1 that $\liminf \beta_n^{-1}\max_{j\leqn|S_j - \tilde{\alpha}_j|>0$. The latter requirement is added to rule out sequences ${\beta_n}$ which grow too fast and entirely overwhelm the fluctuations of $S_n$.

Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n$ for $0<\beta < 1$. This paper proves that $\xi (\beta)$ can be approximated by $\sum_{n=0}^\infty \beta^n Y_n$ under some suitable conditions, where ${Y_n; n \geq 0}$ is a sequence of independent normal random variables. Applications to the law of the iterated logarithm for $\xi(\beta)$ are also discussed.

Let $X_n =\sum_{i=1}^\infty a_i \varepsilon_{n-i}$, where the $\varepsilon_i$ are i.i.d. with mean 0 and finite second moment and the $a_i$ are either summable or regularly varying with index $\in (-1,-1/2)$ . The sequence ${X_n}$ has short memory in the former case and long memory in the latter. For a large class of functions $K$, a new approach is proposed to develop both central ($\sqrt{N}$ rate) and noncentral (non-$\sqrt{N}$ rate) limit theorems for $S_N \equiv \sum_{n=1}^N [K(X_n) - EK (X_n)]$. Specifically, we show that in the short-memory case the central limit theorem holds for $S_N$ and in the long-memory case, $S_N$ can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a non-central limit theorem, respectively. Further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.

In this paper we will investigate the long time behavior of critical branching Brownian motion and (finite variance) super-Brownian motion (the so-called Dawson-Watanabe process) on $\mathbb{R}$^d$. These processes are known to be persistent if $d \geq 3$; that is, there exist nontrivial equilibrium measures. If $d \leq 2$, they cluster; that is, the processes converge to the 0 configuration while the surviving mass piles up in so-called clusters.

We study the spatial profile of the clusters in the “critical” dimension $d = 2$ via multiple space scale analysis. We will also investigate the long-time behavior of these models restricted to finite boxes in $d \geq 2$. On the way, we develop coupling and comparison methods for spatial branching models.

Let $(B(t))_{t \geq 0}$ be the linear Brownian motion starting at 0, and set $X_n (t) =(1/n!)\int_0^t(t-s)^s dB(s)$. Watanabe stated a law of the iterated logarithm for the process $(X_1(t))_{t \geq 0}$, among other things. This paper proposes an elementary proof of this fact, which can be extended to the general case $n\geq 1$. Next, we study the local asymptotic classes (upper and lower) of the $(n +1)$ -dimensional process $U_n =(B, X_1,\ldots,X_n)$ near zero and infinity, and the results obtained are extended to the case where $B$ is the $d$-dimensional Brownian motion.

We derive a large deviation principle for the position at large times $t$ of a one-dimensional annealed Brownian motion in a Poissonian potential in the critical spatial scale $t^{1/3}$. Here “annealed” means that averages are taken with respect to both the path and environment measures. In contrast to the $d$-dimensional case for $d \geq 2$ in the critical scale $t^{d/(d+2)}$ as treated by Sznitman, the rate function which measures the large deviations exhibits three different regimes. These regimes depend on the position of the path at time $t$. Our large deviation principle has a natural application to the study of a one-dimensional annealed Brownian motion with a constant drift in a Poissonian potential.

A Brownian motion in a random Lévy potential V is the informal solution of the stochastic differential equation

$$dX_t = dB_t - 1/2 V'(X_t)dt,$$

where $B$ is a Brownian motion independent of $V$.

We generalize some results of Kawazu-Tanaka, who considered for $V$ a Brownian motion with drift, by proving that $X_t /t$ converges almost surely to a constant, the mean velocity, which we compute in terms of the Lévy exponent $\phi$ of $V$, defined by $\mathbb{E}[e^{mV(t)}]=e{-t\phi(m)}$.

Several stochastic partial differential equations are derived for multidimensional superdiffusions.

Let $X$ be an $\mathbb{R}^d$-valued continuous semimartingale, $T$ a fixed time horizon and $\Theta$ the space of all $\mathbb{R}^d$ -valued predictable $X$ -integrable processes such that the stochastic integral $G(\vartheta)=\int\vartheta dX$ is a square-integrable semimartingale. A recent paper gives necessary and sufficient conditions on $X$ for $G_T(\Theta)$ to be closed in $L^2(P)$. In this paper, we describe the structure of the $L^2$-projection mapping an $\mathscr{F}_T$-measurable random variable $H \in L^2(P)$ on $G_T(\theta)$ and provide the resulting integrand $\vartheta^H \in \Theta$ feedback form. This is related to variance-optimal hedging strategies in financial mathematics and generalizes previous results imposing very restrictive assumptions on $X$. Our proofs use the variance-optimal martingale measure $\tilda{P}$ for $X$ and weighted norm inequalities relating $\tilda{P}$ to the original measure $P$.

We consider inhomogeneous nearest neighbor Bernoulli bond percolation on $mathbb{Z}^d$ where the bonds in a fixed $s$-dimensional hyperplane $1\leq s\leq d-1)$ have density $p_1$ and all other bonds have fixed density, $p_c(\mathbb{Z}^d)$, the homogeneous percolation critical value. For $s\leq 2$, it is natural to conjecture that there is a new critical value, $p_c^s(\mathbb{Z}^d)$ for $p_1$, strictly between $p_c(\mathbb{Z}^d)$ and $p_c(\mathbb{Z}^s)$ ; we prove this for large $d$ and $2 \leq s \leq d-3$. For $s=1$, it is natural to conjecture that $p_c^1(\mathbb{Z}^d) =1$, as shown for $d =2$ by Zhang; we prove this for large $d$. Related results for the contact process are also presented.

We study the ergodic behavior of the contact process on infinite connected graphs of bounded degree. We show that the fundamental notion of complete convergence is not as well behaved as it was thought to be. In particular there are graphs for which complete convergence holds in any number of separated intervals of values of the infection parameter and fails for the other values of this parameter. We then introduce a basic invariant probability measure related to the recurrence properties of the process, and an associated notion of convergence that we call “partial convergence.” This notion is shown to be better behaved than complete convergence, and to hold in certain cases in which complete convergence fails. Relations between partial and complete convergence are presented, as well as tools to verify when these properties hold. For homogeneous graphs we show that whenever recurrence takes place (i.e., whenever local survival occurs) there are exactly two extremal invariant measures.

We study the asymptotic behavior of Brownian motion in steady, unbounded incompressible random flows. We prove an invariance principle for almost all realizations of random flows. The key compactness result is obtained by Moser’s iterative scheme in PDE theory.

The error estimate of the integral kernel for the Trotter product formula for Schrödinger operators is shown. A basic tool for doing so is the Feynman-Kac formula based on the pinned Brownian motion. This formula enables us to express the integral kernel in handleable form and hence estimate it. As a consequence the Trotter product formula in the $L_p$ operator norm is obtained.

The asymptotic behaviors of the principal eigenvalue and the corresponding normalized eigenfunction of the operator $G^\varepsilon f = (\varepsilon/2)\triangle f + g \triangledown f +(l/\varepsilon)f$ for small $\varepsilon$ are studied. Under some conditions, the first order expansions for them are obtained. Two applications to risk-sensitive control problems are also mentioned.

By using logarithmic transformations, an explicit lower bound estimate of heat kernels is obtained for diffusion processes on Riemannian manifolds. This estimate is sharp for both short and long times, especially for heat kernels on a compact manifold, and is extended to manifolds with unbounded curvature.

We show that increments of size $h_n$ from the uniform quantile and n ?.uniform empirical processes in the neighborhood of a fixed point $t_0 \in (0,1)$ may have different rates of almost sure convergence to 0 in the range where $h_n \to 0$ and $nh_n /\log n \to \infty$. In particular, when $h_n = n^{-\lambda}$ with $0<\lambda<1$, we obtain that these rates are identical for $1/2<\lambda<1$, and distinct for $0<\lambda<1/2$. This phenomenon is shown to be a consequence of functional laws of the iterated logarithm for local quantile processes, which we describe in a more general setting. As a consequence of these results, we prove that, for any $\varaepsilon>0$, the best possible uniform almost sure rate of approximation of the uniform quantile process by a normed Kiefer process is not better than $O(n ^{-1/4}\log n)^{-\varepsilon})$.

In this paper we provide sharp bounds on the $L_p$-norms of randomly stopped $U$-statistics. These bounds consist mainly of decoupling inequalities designed to reduce the level of dependence between the $U$-statistics and the stopping time involved. We apply our results to obtain Wald’s equation for $U$-statistics, moment convergence theorems and asymptotic expansions for the moments of randomly stopped $U$-statistics. The proofs are based on decoupling inequalities, symmetrization techniques, the use of subsequences and induction arguments.