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: $$ Q_{S}(\varepsilon):= \sup_{x \in\mathbb{R}}\mathbb{P} (x < S \leq x+\varepsilon) $$ where $S$ is a weighted sum of noncentral chi-square random variables: $$ S:= \sum_{k=1}^\infty λ_k (Z_k^2 - 1) + μ_kZ_k $$ Here, $\{Z_k\}_{k=1}^\infty$ is a sequence of independent standard Gaussian random variables and $\{λ_k\}_{k=1}^\infty, \{μ_k\}_{k=1}^\infty$ 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 $\|λ\|_2$ is negligible compared to $\|μ\|_2$ and chi-square estimates if $\|μ\|_{2}$ is negligible compared to $\|λ\|_2$. Our bound generalizes existing bounds in various ways. In particular, we make no assumptions regarding the number of nonzero $|λ_k|$ or the size of the minimal $|λ_k|$, nor do we make any assumptions on the signs of $λ_k$. Finally, we apply our bound to some examples of interest, specifically quadratic forms that arise in limit theorems for second-order U-statistics.