Using multiple stochastic integrals and the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for a specific non-Gaussian self-similar process, the Rosenblatt process. We apply our results to the design of strongly consistent statistical estimators for the self-similarity parameter H. Although, in the case of the Rosenblatt process, our estimator has non-Gaussian asymptotics for all H>1/2, we show the remarkable fact that the process’s data at time 1 can be used to construct a distinct, compensated estimator with Gaussian asymptotics for H∈(1/2, 2/3).

For arrays (S_i,j)_1≤i≤j of random variables that are stationary in an appropriate sense, we show that the fluctuations of the process (S_1,n)_n=1^∞ can be bounded in terms of a measure of the “mean subadditivity” of the process (S_i,j)_1≤i≤j. We derive universal upcrossing inequalities with exponential decay for Kingman’s subadditive ergodic theorem, the Shannon–MacMillan–Breiman theorem and for the convergence of the Kolmogorov complexity of a stationary sample.

In this paper we establish a multivariate exchangeable pairs approach within the framework of Stein’s method to assess distributional distances to potentially singular multivariate normal distributions. By extending the statistics into a higher-dimensional space, we also propose an embedding method which allows for a normal approximation even when the corresponding statistics of interest do not lend themselves easily to Stein’s exchangeable pairs approach. To illustrate the method, we provide the examples of runs on the line as well as double-indexed permutation statistics.

Let X₁, …, X_n be i.i.d. random observations. Let ${\mathbb{S}=\mathbb{L}+\mathbb{T}}$ be a U-statistic of order k≥2 where $\mathbb{L}$ is a linear statistic having asymptotic normal distribution, and $\mathbb {T}$ is a stochastically smaller statistic. We show that the rate of convergence to normality for $\mathbb{S}$ can be simply expressed as the rate of convergence to normality for the linear part $\mathbb{L}$ plus a correction term, $(\operatorname{var}\mathbb{T})\ln^{2}(\operatorname{var}\mathbb{T})$, under the condition ${\mathbb{E}\mathbb{T}^{2}<\infty}$. An optimal bound without this log factor is obtained under a lower moment assumption ${\mathbb{E}|\mathbb{T}|^{\alpha}< \infty}$ for ${\alpha < 2}$. Some other related results are also obtained in the paper. Our results extend, refine and yield a number of related-known results in the literature.

We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion B with Hurst index H=1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to B.

We show how to detect optimal Berry–Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein’s method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (ii) to “exploding” quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer–Major CLT for functionals of a fractional Brownian motion.

This article presents a method for finding the critical probability p_c for the Bernoulli bond percolation on graphs with the so-called tree-like structure. Such a graph can be decomposed into a tree of pieces, each of which has finitely many isomorphism classes. This class of graphs includes the Cayley graphs of amalgamated products, HNN extensions or general groups acting on trees. It also includes all transitive graphs with more than one end.

The idea of the method is to find a multi-type Galton–Watson branching process (with a parameter p) which has finite expected population size if and only if the expected percolation cluster size is finite. This provides sufficient information about p_c. In particular, if the pairwise intersections of pieces are finite, then p_c is the smallest positive p such that det(M−1)=0, where M is the first-moment matrix of the branching process. If the pieces of the tree-like structure are finite, then p_c is an algebraic number and we give an algorithm computing p_c as a root of some algebraic function.

We show that any Cayley graph of a virtually free group (i.e., a group acting on a tree with finite vertex stabilizers) with respect to any finite generating set has a tree-like structure with finite pieces. In particular, we show how to compute p_c for the Cayley graph of a free group with respect to any finite generating set.

We study invasion percolation in two dimensions. We compare connectivity properties of the origin’s invaded region to those of (a) the critical percolation cluster of the origin and (b) the incipient infinite cluster. To exhibit similarities, we show that for any k≥1, the k-point function of the first so-called pond has the same asymptotic behavior as the probability that k points are in the critical cluster of the origin. More prominent, though, are the differences. We show that there are infinitely many ponds that contain many large disjoint p_c-open clusters. Further, for k>1, we compute the exact decay rate of the distribution of the radius of the kth pond and see that it differs from that of the radius of the critical cluster of the origin. We finish by showing that the invasion percolation measure and the incipient infinite cluster measure are mutually singular.

If we consider the contact process with infection rate λ on a random graph on n vertices with power law degree distributions, mean field calculations suggest that the critical value λ_c of the infection rate is positive if the power α>3. Physicists seem to regard this as an established fact, since the result has recently been generalized to bipartite graphs by Gómez-Gardeñes et al. [Proc. Natl. Acad. Sci. USA 105 (2008) 1399–1404]. Here, we show that the critical value λ_c is zero for any value of α>3, and the contact process starting from all vertices infected, with a probability tending to 1 as n→∞, maintains a positive density of infected sites for time at least exp(n^1−δ) for any δ>0. Using the last result, together with the contact process duality, we can establish the existence of a quasi-stationary distribution in which a randomly chosen vertex is occupied with probability ρ(λ). It is expected that ρ(λ)∼Cλ^β as λ→0. Here we show that α−1≤β≤2α−3, and so β>2 for α>3. Thus even though the graph is locally tree-like, β does not take the mean field critical value β=1.

We consider the weakly asymmetric exclusion process on a bounded interval with particle reservoirs at the endpoints. The hydrodynamic limit for the empirical density, obtained in the diffusive scaling, is given by the viscous Burgers equation with Dirichlet boundary conditions. We prove the associated dynamical large deviations principle.

In this paper we introduce the notion of fractional martingale as the fractional derivative of order α of a continuous local martingale, where α∈(−½, ½), and we show that it has a nonzero finite variation of order 2/(1+2α), under some integrability assumptions on the quadratic variation of the local martingale. As an application we establish an extension of Lévy’s characterization theorem for the fractional Brownian motion.

We prove that the total range of Super-Brownian motion with quadratic branching mechanism has an exact packing measure with respect to the gauge function g(r)=r⁴(log log 1/r)⁻³ in super-critical dimensions d≥5. More precisely, we prove that the total occupation measure of Super-Brownian motion is equal to the g-packing measure restricted to its range, up to a deterministic multiplicative constant that only depends on space dimension d.

In this article we consider a Brownian motion with drift of the form

dS_t=μ_t dt+dB_t for t≥0,

with a specific nontrivial (μ_t)_t≥0, predictable with respect to $\mathbb{F}^{B}$, the natural filtration of the Brownian motion B=(B_t)_t≥0. We construct a process H=(H_t)_t≥0, also predictable with respect to $\mathbb{F}^{B}$, such that ((H⋅S)_t)_t≥0 is a Brownian motion in its own filtration. Furthermore, for any δ>0, we refine this construction such that the drift (μ_t)_t≥0 only takes values in ]μ−δ, μ+δ[, for fixed μ>0.

The aim of this paper is to give a characterization of the dimension free concentration of measure phenomenon in terms of transportation-cost inequalities. We apply this theorem to give a new and very short proof of a result by Otto and Villani. Another application is to show that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of all these results rely on simple Large Deviations techniques.