We consider the following problem:given an i.i.d. family of Bernoulli random variables indexed by $\mathbb{Z}^d$, find a random occupied site $X \in \mathbb{Z}^d$ such that relative to $X$, the other random variables are still i.i.d. Bernoulli. Results of Thorisson imply that such an $X$ exists for all $d$. Liggett proved that for$d = 1$, there exists an $X$ with tails $P(|X|\geq t)$ of order $ct^(-1 /2}$, but none with finite $1/2$th moment. We prove that for general $d$ there exists a solution with tails of order $ct^{-d/2}$, while for $d = 2$ there is none with finite first moment. We also prove analogous results for a continuum version of the same problem. Finally we prove a result which strongly suggests that the tail behavior mentioned above is the best possible for all$d$.

We formulate effective versions of the Borel–Cantelli lemmas using a coefficient from Kolmogorov complexity. We then use these effective versions to lift the effective content of the law of large numbers and the law of the iterated logarithm.

Let $X_1\ldots,X_n, n \geq 1$, be independent, identically distributed random variables and consider the Student $t$-statistic $T_n$ based upon these random variables. Giné, Götze and Mason (1997) proved that $T_n$ converges in distribution to a standard normal random variable if and only if $X$ is in the domain of attraction of a normal random variable and$EX = 0$. We shall show that roughly the same holds true for the bootstrapped Student $t$- statistic $T_n^*$. In the process we shall disclose all the possible subsequential limiting laws of $T_^*n$. The proofs introduce a number of amusing tricks that may be of independent interest.

We identify the possible cluster sets for a general lawof the iterated logarithm in the Banach space setting,and showthat all the possible limit sets arise as cluster sets for some random vector in an arbitrary separable Banach space. This extends previous results obtained in .nite dimensional Euclidean spaces.

We investigate the set of invariant probability distributions for measure-valued diffusion processes corresponding to semilinear operators of the form $u_t = L_0 u + \beta u - \alpha u^2$, where $L_0 = 1/2 \sum_{i,j=1}^d a_{i,j} \frac{\partial^2}{\partial x_i \partial x_j}+ \sum_{i=1}^d b_1\frac{\partial}{\partial x_i}$

Suppose $X = (X_x)_{x\in\mathbb{Z}^d}$ is a white noise process, and $H (B)$, defined for finite subsets $B$ of $\math{Z}^d$, is determined in a stationary way by the restriction of $X$ to $B$. Using a martingale approach, we prove a central limit theorem (CLT) for $H$ as $B$ becomes large, subject to $H$ satisfying a “stabilization” condition (the effect of changing $X _x$ at a single site needs to be local). This CLT is then applied to component counts for percolation and Boolean models, to the size of the big cluster for percolation on a box, and to the final size of a spatial epidemic.

We consider a class of haploid population models with nonoverlapping generations and fixed population size $N$ assuming that the family sizes within a generation are exchangeable random variables. A weak convergence criterion is established for a properly scaled ancestral process as $N \to \infty$. It results in a full classification of the coalescent generators in the case of exchangeable reproduction. In general the coalescent process allows for simultaneous multiple mergers of ancestral lines.

We show that the branching random walk on a Galton–Watson tree may have one or two phase transitions, depending on the relative sizes of the mean degree and the maximum degree. We show that there are some Galton–Watson trees on which the branching random walk has one phase transition while the contact process has two; this contradicts a conjecture of Madras and Schinazi. We show that the contact process has only one phase transition on some trees of uniformly exponential growth and bounded degree, contradicting a conjecture of Pemantle.

We define a class of processes on the boundary of a regular tree that can be viewed as “stable” Lévy processes on $(\mathbb{Z}/n_0\mathbb{Z})^\mathbb{N}$. We show that the range of these processes can be compared with a Bernoulli percolation as in Peres which easily leads to various results on the intersection properties. We develop an alternative approach based on the comparison with a branching random walk. By this method we establish the existence of points of in finite multiplicity when the index of the process equals the dimension of the state space, as for planar Brownian motion.

Improper regular conditional distributions (rcd’s) given a $\sigma$-field $\mathscr{A}$ have the following anomalous property. For sets $A \in \mathscr{A}, \mathrm{Pr}(A|\mathscr{A})$ is not always equal to the indicator of $A$. Such a property makes the conditional probability puzzling as a representation of uncertainty. When rcd’s exist and the$\sigma$-field $\mathscr{A}$ is countably generated, then almost surely the rcd is proper. We give sufficient conditions for an rcd to be improper in a maximal sense, and show that these conditions apply to the tail $\sigma$-field and the $\sigma$-field of symmetric events.

Let $F$ be any distribution function on $\mathbb{R}$, and $F_n$ be the$n$th empirical distribution function based on fariables i.i.d. ($F$). It is shown that for $2<p<\infty$ and a constant $C(p) < \infty$, not depending on $F$, on some probability space there exist $F_n$ and Brownian bridges $B_n$ such that for the Wiener-Young $p$-variation norm $\|\cdot\|_{[p]}, E\|n^{1/2}(F_n - F) - B_n \circ F\|_{[p]} \leq C(p)n^{(2-p)/(2p)}$, where $(B_n \circ F)(x))$. The expectation can be replaced by an Orlicz norm of exponential order. Conversely, if $F$ is continuous, then for any stochastic process $V(t,\omega)$ continuous in $t$ for almost all $\omega$, such as $B_n \circ F$, summation over $n$ distinct jumps shows that $\|n^{1/2}(F_n - F) - V\|_{[p]} \geq n^{(2-p)/(2p)}$, so the upper bound in expectation is best possible up to the constant $C(p)$. In the proof, $B_n$ is linked to $F_n$ by the Komlós, Major and Tusnády construction, as for the supremum norm $(p = \infty)$.

Suppose that a coin with bias $\theta$ is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let $\mu_\theta$ be the distribution ofthe observed sequence of coin tosses, and let $u_n$ denote the chance of a renewal at time $n$. Harris and Keane showed that if $\sum_{n=1}^\infty u_n^2 = \infty$ then $\mu_\theta$ and $\mu_0$ are singular, while if $\sum_{n=1}^\infty u_n^2 < \infty$ and $\theta$ is small enough, then $\mu_\theta$ is absolutely continuous with respect to$\mu_0$. They conjectured that absolute continuity should not depend on $\theta$, but only on the square-summability of $\{u_n\}$. We show that in fact the power law governing the decay of $\{u_n\}$ is crucial, and for some renewal sequences $\{u_n\}$, there is a phase transition at a critical parameter $\theta_c \in (0,1):$ for $|\theta| < \theta_c$ the measures $\mu_\theta$ and $\mu_0$ are mutually absolutely continuous, but for $|\theta| > \theta_c$, they are singular. We also prove that when $u_n=O(n ^{-1}), the measures $\mu_\theta$ for $\theta \in [-1,1] are all mutually absolutely continuous.

We consider the lattice version of the free field in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall. The basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure.

We define a local time flow of skew Brownian motions ,that is, a family of solutions to the stochastic differential equation defining the skew Brownian motion, starting from different points but driven by the same Brownian motion. We prove several results on distributional and path properties of the flow. Our main result is a version of the Ray–Knight theorem on local times. In our case, however, the local time process viewed as a function of the spatial variable is a pure jump Markov process rather than a diffusion.

We solve the problem posed by S.A. Kalikow whether the event that the $x$-coordinate of a random walk in a two-dimensional random environment approaches $\infty$ has necessarily probability either zero or one. The answer is yes if we assume the environment to be i.i.d.and in general no if we allow the environment to be just stationary and ergodic.

We study quasi-stationary measures for conservative particle systems in the in finite lattice. Existence of quasi-stationary measures is established for a fairly general class of reversible systems. For the special cases of a system of independent random walks and the symmetric simple exclusion process, it is shown that qualitative features of quasi-stationary measures change drastically with dimension.

We describe a Vervaat-like path transformation for the reflected Brownian bridge conditioned on its local time at 0: up to random shifts, this process equals the two processes constructed froma Brownian bridge and a Brownian excursion by adding a drift and then taking the excursions over the current minimum. As a consequence, these three processes have the same occupation measure, which is easily found.

The three processes arise as limits, in three different ways, of profiles associated to hashing with linear probing, or, equivalently, to parking functions.

We introduce a class of interesting stochastic processes based on Brownian-time processes. These are obtained by taking Markov processes and replacing the time parameter with the modulus of Brownian motion. They generalize the iterated Brownian motion (IBM) of Burdzy and the Markov snake of Le Gall, and they introduce new interesting examples. After defining Brownian-time processes, we relate them to fourth order parabolic partial differential equations (PDE’s). We then study their exit problem as they exit nice domains in $\mathbb{R}^d$ , and connect it to elliptic PDE’s. We show that these processes have the peculiar property that they solve fourth order parabolic PDE’s, but their exit distribution—at least in the standard Brownian time process case—solves the usual second order Dirichlet problem. We recover fourth order PDE’s in the elliptic setting by encoding the iterative nature of the Brownian-time process, through its exit time, in a standard Brownian motion. We also show that it is possible to assign a formal generator to these non-Markovian processes by giving such a generator in the half-derivative sense.

The existence of a martingale solution to 2-dimensional stochastic Euler equations is proved. The constructed solution is a limit as the viscosity converges to zero of a sequence of solutions to modified Navier–Stokes equations.

Superprocesses (under the name continuous state branchingprocesses) appeared, first, in a pioneering work of S.Watanabe [J. Math. Kyoto Univ. 8 (1968)141 –167 ]. Deep results on paths of the super-Brownian motion were obtained by Dawson, Perkins, Le Gall and others.

In earlier papers, a superprocess was interpreted as a Markov process $X_t$ in the space of measures. This is not sufficient for a probabilistic approach to boundary value problems. A reacher model based on the concept of exit measures was introduced by E.B.Dynkin [Probab. Theory Related Fields 89 (1991) 89 –115 ]. A model of a superprocess as a system of exit measures from time-space open sets was systematically developed in 1993 [E.B. Dynkin, Ann.Probab. 21 (1993)1185 –1262 ]. In particular, branchingand Markov properties of such a system were established and used to investigate partial differential equations. In the present paper, we show that the entire theory of superprocesses can be deduced from these properties.

On généralise le théoème de la limite centrale presque-sûre aux martingales fonctionnelles additives d’un processus de Markov récurrent. Et dans le cas de la récurrence positive, on dégage deux principes d’invariance analogues au théorème de la limite centrale et àla loi dulogarithme itéré.

We extend the almost-sure central limit theorem to martingale additive functionals of a recurrent Markov process.In the particular case of positive recurrence, we also derive two invariance principles similar to the central limit theorem and the law of the iterated logarithm.