Consider routing traffic on the N×N torus, simultaneously between all source-destination pairs, to minimize the cost ∑_ec(e)f²(e), where f(e) is the volume of flow across edge e and the c(e) form an i.i.d. random environment. We prove existence of a rescaled N→∞ limit constant for minimum cost, by comparison with an appropriate analogous problem about minimum-cost flows across a M×M subsquare of the lattice.

We consider the parabolic Anderson problem ∂_tu=Δu+ξ(x)u on ℝ₊×ℤ^d with localized initial condition u(0, x)=δ₀(x) and random i.i.d. potential ξ. Under the assumption that the distribution of ξ(0) has a double-exponential, or slightly heavier, tail, we prove the following geometric characterization of intermittency: with probability one, as t→∞, the overwhelming contribution to the total mass ∑_xu(t, x) comes from a slowly increasing number of “islands” which are located far from each other. These “islands” are local regions of those high exceedances of the field ξ in a box of side length 2t log²t for which the (local) principal Dirichlet eigenvalue of the random operator Δ+ξ is close to the top of the spectrum in the box. We also prove that the shape of ξ in these regions is nonrandom and that u(t, ⋅) is close to the corresponding positive eigenfunction. This is the geometric picture suggested by localization theory for the Anderson Hamiltonian.

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, α, β)-branching particle system [particles moving in ℝ^d according to a symmetric α-stable Lévy process, branching law in the domain of attraction of a (1+β)-stable law, 0<β<1, uniform Poisson initial state] in the case of intermediate dimensions, α/β<d<α(1+β)/β. The limit is a process of the form Kλξ, where K is a constant, λ is the Lebesgue measure on ℝ^d, and ξ=(ξ_t)_t≥0 is a (1+β)-stable process which has long range dependence. For α<2, there are two long range dependence regimes, one for β>d/(d+α), which coincides with the case of finite variance branching (β=1), and another one for β≤d/(d+α), where the long range dependence depends on the value of β. The long range dependence is characterized by a dependence exponent κ which describes the asymptotic behavior of the codifference of increments of ξ on intervals far apart, and which is d/α for the first case (and for α=2) and (1+β−d/(d+α))d/α for the second one. The convergence proofs use techniques of $\mathscr{S}'(\mathbb {R}^{d})$-valued processes.

This paper studies the limits of a spatial random field generated by uniformly scattered random sets, as the density λ of the sets grows to infinity and the mean volume ρ of the sets tends to zero. Assuming that the volume distribution has a regularly varying tail with infinite variance, we show that the centered and renormalized random field can have three different limits, depending on the relative speed at which λ and ρ are scaled. If λ grows much faster than ρ shrinks, the limit is Gaussian with long-range dependence, while in the opposite case, the limit is independently scattered with infinite second moments. In a special intermediate scaling regime, there exists a nontrivial limiting random field that is not stable.

We show that a stationary IDp process (i.e., an infinitely divisible stationary process without Gaussian part) can be written as the independent sum of four stationary IDp processes, each of them belonging to a different class characterized by its Lévy measure. The ergodic properties of each class are, respectively, nonergodicity, weak mixing, mixing of all order and Bernoullicity. To obtain these results, we use the representation of an IDp process as an integral with respect to a Poisson measure, which, more generally, has led us to study basic ergodic properties of these objects.

We show that the largest disc covered by a simple random walk (SRW) on ℤ² after n steps has radius n^1/4+o(1), thus resolving an open problem of Révész [Random Walk in Random and Non-Random Environments (1990) World Scientific, Teaneck, NJ]. For any fixed ℓ, the largest disc completely covered at least ℓ times by the SRW also has radius n^1/4+o(1). However, the largest disc completely covered by each of ℓ independent simple random walks on ℤ² after n steps is only of radius $n^{1/(2+2\sqrt{\ell})+o(1)}$. We complement this by showing that the radius of the largest disc completely covered at least a fixed fraction α of the maximum number of visits to any site during the first n steps of the SRW on ℤ², is $n^{(1-\sqrt{\alpha})/4+o(1)}$. We also show that almost surely, for infinitely many values of n it takes about n^1/2+o(1) steps after step n for the SRW to reach the first previously unvisited site (and the exponent 1/2 is sharp). This resolves a problem raised by Révész [Ann. Probab. 21 (1993) 318–328].

We study the upper tail behaviors of the local times of the additive stable processes. Let X₁(t), …, X_p(t) be independent, d-dimensional symmetric stable processes with stable index 0<α≤2 and consider the additive stable process X̅(t₁, …, t_p)=X₁(t₁)+⋯+X_p(t_p). Under the condition d<αp, we obtain a precise form of the large deviation principle for the local time

η^x([0, t]^p)=∫₀^t⋯∫₀^tδ_x(X₁(s₁)+⋯+X_p(s_p)) ds₁⋯ ds_p

of the multiparameter process X̅(t₁, …, t_p), and for its supremum norm sup _x∈ℝ^dη^x([0, t]^p). Our results apply to the law of the iterated logarithm and our approach is based on Fourier analysis, moment computation and time exponentiation.

Let $(Ω, ℬ, P)$ be a probability space, $\mathscr{A}\subset\mathscr{B}$ a sub-$σ$-field, and $μ$ a regular conditional distribution for $P$ given $\mathscr{A}$. Necessary and sufficient conditions for $μ(ω)(A)$ to be 0–1, for all $A\in\mathscr{A}$ and $ω∈A_0$, where $A_{0}\in\mathscr{A}$ and $P(A_0)=1$, are given. Such conditions apply, in particular, when $\mathscr{A}$ is a tail sub-$σ$-field. Let $H(ω)$ denote the $\mathscr{A}$-atom including the point $ω∈Ω$. Necessary and sufficient conditions for $μ(ω)(H(ω))$ to be 0–1, for all $ω∈A_0$, are also given. If $(Ω, ℬ)$ is a standard space, the latter 0–1 law is true for various classically interesting sub-$σ$-fields $\mathscr{A}$, including tail, symmetric, invariant, as well as some sub-$σ$-fields connected with continuous time processes.

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n×p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance Σ_p. We show that for a large class of covariance matrices Σ_p, the largest eigenvalue of X^*X is asymptotically distributed (after recentering and rescaling) as the Tracy–Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p.

The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

Nous étudions les chaînes de Markov (X_n)_n∈Z gouvernées par une relation de récurrence de la forme X_n+1=f(X_n, V_n+1), où (V_n)_n∈Z est une suite de variables aléatoires indépendantes et de même loi telle pour tout n∈Z, V_n+1 est indépendante de la suite ((X_k, V_k))_k≤n. L’objet de l’article est de donner une condition nécessaire et suffisante pour que les innovations (V_n)_n∈Z déterminent complètement la suite (X_n)_n∈Z et de décrire l’information manquante dans le cas contraire.

We show that the edges crossed by a random walk in a network form a recurrent graph a.s. In fact, the same is true when those edges are weighted by the number of crossings.

We develop the mathematics of a filtration shrinkage model that has recently been considered in the credit risk modeling literature. Given a finite collection of points x₁<⋯<x_N in ℝ, the region indicator function R(x) assumes the value i if x∈(x_i−1, x_i]. We take $\mathbb{F}$ to be the filtration generated by (R(X_t))_t≥0, where X is a diffusion with infinitesimal generator $\mathscr{A}$. We prove a martingale representation theorem for $\mathbb{F}$ in terms of stochastic integrals with respect to N random measures whose compensators have a simple form given in terms of certain Lévy measures F^j±_i, which are related to the differential equation $\mathscr{A}u=\lambda u$.

Let the random variable Z_n,k denote the number of increasing subsequences of length k in a random permutation from S_n, the symmetric group of permutations of {1, …, n}. In a recent paper [Random Structures Algorithms 29 (2006) 277–295] we showed that the weak law of large numbers holds for Z_n,kn if k_n=o(n^2/5); that is, $$\lim_{n\to\infty}\frac{Z_{n,k_{n}}}{EZ_{n,k_{n}}}=1\qquad \mbox{in probability}.$$ The method of proof employed there used the second moment method and demonstrated that this method cannot work if the condition k_n=o(n^2/5) does not hold. It follows from results concerning the longest increasing subsequence of a random permutation that the law of large numbers cannot hold for Z_n,kn if k_n≥cn^1/2, with c>2. Presumably there is a critical exponent l₀ such that the law of large numbers holds if k_n=O(n^l), with l<l₀, and does not hold if $\limsup_{n\to\infty}\frac{k_{n}}{n^{l}}>0$, for some l>l₀. Several phase transitions concerning increasing subsequences occur at l=½, and these would suggest that l₀=½. However, in this paper, we show that the law of large numbers fails for Z_n,kn if $\limsup_{n\to\infty}\frac{k_{n}}{n^{4/9}}=\infty$. Thus, the critical exponent, if it exists, must satisfy $l_{0}\in[\frac{2}{5},\frac{4}{9}]$.

Let X_t be any additive process in ℝ^d. There are finite indices δ_i, β_i, i=1, 2 and a function u, all of which are defined in terms of the characteristics of X_t, such that $$\begin{aligned}\liminf_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\delta_{1},\cr\infty,\quad \mbox{if }\eta < \delta_{2},}\\ \limsup_{t\rightarrow0}u(t)^{-1/\eta}X_{t}^{*}=\cases{0,\phantom{\infty,\quad\!\!\!\!\!\!}\mbox{if }\eta>\beta_{2},\cr\infty,\quad \mbox{if }\eta < \beta_{1},}\end{aligned}\qquad \mbox{a.s.},$$ where X_t^*=sup _0≤s≤t|X_s|. When X_t is a Lévy process with X₀=0, δ₁=δ₂, β₁=β₂ and u(t)=t. This is a special case obtained by Pruitt. When X_t is not a Lévy process, its characteristics are complicated functions of t. However, there are interesting conditions under which u becomes sharp to achieve δ₁=δ₂, β₁=β₂.