U-statistic central limit theorem ($\sqrt{n}(U_n-\eta )\Rightarrow N(0,m^2{\zeta }_1)$)

For an iid sample and symmetric square-integrable kernel $h$ of fixed order $m$, put $\eta =\mathbb{E}h(X_1,\dots ,X_m)$ and $h_1(x)=\mathbb{E}[h(x,X_2,\dots ,X_m)]-\eta$. If ${\zeta }_1=\mathrm{Var}(h_1(X_1))>0$, the displayed limit holds. The first Hoeffding projection is $(m/n){\sum }_i{h}_1(X_i)$; 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.

The displayed sum is the first-order projection of a centred U-statistic onto sums of functions of individual observations. Its summands are independent and have mean zero. Subtracting it leaves a remainder with conditional expectation zero in each individual observation, a useful decomposition for variance and normal-limit calculations.