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:
$$
S:= \sum_{k=1}^\infty \lambda_k (Z_k^2 - 1) + \mu_kZ_k
$$
Here, $\{Z_k\}_{k=1}^\infty$ is a sequence of independent standard Gaussian random variables and $\{\lambda_k\}_{k=1}^\infty, \{\mu_k\}_{k=1}^\infty$ 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 $\|\lambda\|_2$ is negligible compared to $\|\mu\|_2$ and chi-square estimates if $\|\mu\|_{2}$ is negligible compared to $\|\lambda\|_2$. Our bound generalizes existing bounds in various ways. In particular, we make no assumptions regarding the number of nonzero $|\lambda_k|$ or the size of the minimal $|\lambda_k|$, nor do we make any assumptions on the signs of $\lambda_k$. Finally, we apply our bound to some examples of interest, specifically quadratic forms that arise in limit theorems for second-order U-statistics.
From: Abhimanyu Choudhary [view email]
[v1]
Wed, 24 Jun 2026 06:09:03 UTC (40 KB)
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