Limit distributions of conditionalU-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 ofU ^h_n (t), the so-calledconditional U-statistics, introduced by Stute.⁽¹⁰⁾ They are estimators of functions of the formm ^h(t)=E{h(Y ₁,...,Y _k )|X ₁=t ₁,...,X _k =t _k },t=(t ₁,...,t _k )∈ℝ^k whereE |h|<∞. Heret is fixed. In caset ₁=...=t_k=t (say), we describe the limiting random variables asmultiple Wiener integrals with respect toP _t, the conditional distribution ofY, givenX=t. Whent _i, 1⩽i⩽k, are not all equal, we introduce and use a slightly generalized version of a multiple Wiener integral.