Limit distributions of conditional U-statistics

Abstract

Let{(X_n, Y_n)}_{n ≥ 1} be a sequence of i.i.d. bi-variate vectors. In this article, we study the possible limit distributions of U^h _n (t), the so-called conditional U-statistics, introduced by Stute. ⁽¹⁰⁾ They are estimators of functions of the form m^h(t)=E{h(Y ₁,...,Y_k)|X₁=t _l,...,X_k = t_k},t=(t ₁,...,t_k) ∈R^k where E|h|<∞. Here t is fixed. In case t_l=...=t_k=t (say), we describe the limiting random variables as multiple Wiener integrals with respect to P_t, the conditional distribution of Y, given X=t. When t_i, 1≤i≤k, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.