Abstract
Let (Sn)n≥0 be a Z-random walk and (ξx)x,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 R2 with values in R. We study the weak convergence of the sequence Un,n ∈ N, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by Σ h(ξsi,ξsj)t∈ [0,1]. Statistical applications are presented, in particular we prove a strong law of large numbers for U-statistics indexed by a one-dimensional random walk using a result of [1].