Limit theorems for U-statistics indexed by a one dimensional random walk

Abstract

Let (Sn)n≥0 be a -random walk and be a sequence of independent and identically distributed -valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on with values in . We study the weak convergence of the sequence , with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by 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].