Let ${\left(S_n\right)}_{n\ge 0}$ be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let $h$ be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in $D[0,1]$ the set of right continuous real-valued functions with left limits, defined by$$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$Statistical applications are presented, in particular we prove a strong law of large numbers for $U$-statistics indexed by a one-dimensional...

Let (Sn)n≥0 be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by $$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$ Statistical applications are presented, in particular we prove a strong law of large numbers for U-statistics indexed by...

Random interlacements at level $u$ is a one parameter family of connected random subsets of ${\mathbb{Z}}^d$, $d\ge 3$ (Ann. Math.171(2010) 2039–2087). Its complement, the vacant set at level $u$, exhibits a non-trivial percolation phase transition in $u$ (Comm. Pure Appl. Math.62 (2009) 831–858; Ann. Math.171 (2010) 2039–2087), and the infinite connected component, when it exists, is almost surely unique (Ann. Appl. Probab.19(2009) 454–466). In this paper we study local percolative properties of the vacant set of random interlacements...

Let be a locally compact non-compact metric group. Assuming that is abelian we construct symmetric aperiodic random walks on with probabilities $n\mapsto \mathbb{P}\left(S_{2n}\in V\right)$ of return to any neighborhood V of the neutral element decaying at infinity almost as fast as the exponential function n ↦ exp(-n). We also show that for some discrete groups , the decay of the function $n\mapsto \mathbb{P}\left(S_{2n}\in V\right)$ can be made as slow as possible by choosing appropriate aperiodic random walks Sₙ on .