On the L{é}vy concentration function of Gaussian quadratic forms with applications to second order U-statistics

Abstract

We provide an upper-bound for the L{é}vy concentration function: $$Q_S(\epsilon ):=\underset{x\in \mathbb{R}}{\sup }\mathbb{P}(x<S\le x+\epsilon )$$ where $S$ is a weighted sum of noncentral chi-square random variables: $$S:=\sum _{k=1}^{\infty }\hat{I}{»}_k(Z_k^2-1)+\hat{I}{¼}_k{Z}_k$$ Here, $\{Z_k{\}}_{k=1}^{\infty }$ is a sequence of independent standard Gaussian random variables and $\{\hat{I}{»}_k{\}}_{k=1}^{\infty },\{\hat{I}{¼}_k{\}}_{k=1}^{\infty }$ are real valued, square summable sequences. Random variables of this type often appear as limiting distributions of second order U-statistics. Our bound is adaptive, in that it recovers (up to constant factors) Gaussian type concentration function estimates if $\Vert \hat{I}»{\Vert }_2$ is negligible compared to $\Vert \hat{I}¼{\Vert }_2$ and chi-square estimates if $\Vert \hat{I}¼{\Vert }_2$ is negligible compared to $\Vert \hat{I}»{\Vert }_2$. Our bound generalizes existing bounds in various ways. In particular, we make no assumptions regarding the number of nonzero $|\hat{I}{»}_k|$ or the size of the minimal $|\hat{I}{»}_k|$, nor do we make any assumptions on the signs of $\hat{I}{»}_k$. Finally, we apply our bound to some examples of interest, specifically quadratic forms that arise in limit theorems for second-order U-statistics.