A Remark on Convergence in Distribution of $U$-Statistics

Abstract

It is proved that, for $h$ measurable and symmetric in its arguments and $X_i$ i.i.d., if the sequence $\{n^{-m/2} \sum_{i_1,\ldots,i_m\leq n,i_j\neq i_k \text{if} j\neq k} h(X_{i_1},\ldots, X_{i_m})\}^\infty_{n=1}$ is stochastically bounded, then $Eh^2 < \infty$ and $Eh(X_1,x_2,\ldots,x_m) = 0$ a.s.