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

Guillotin-Plantard, Nadine; Ladret, Véronique

ESAIM: Probability and Statistics, Tome 9 (2005), pp. 98-115.

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 random walk using a result of [1].

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