= U-statistic central limit theorem
{c}
{title2=$\sqrt n(U_n-\eta)\Rightarrow N(0,m^2\zeta_1)$}
For an iid sample and symmetric square-integrable kernel $h$ of fixed order $m$, put $\eta=\mathbb Eh(X_1,\ldots,X_m)$ and $h_1(x)=\mathbb E[h(x,X_2,\ldots,X_m)]-\eta$. If $\zeta_1=\operatorname{Var}(h_1(X_1))>0$, the displayed limit holds. The first <Hoeffding projection> is $(m/n)\sum_ih_1(X_i)$; its orthogonal remainder is $O_{L^2}(n^{-1})$, giving the limit by the ordinary <central limit theorem>. Degenerate kernels require a different limiting theory.
Back to article page
