[Paper Review] Two-sided inequalities for the density function's maximum of weighted sum of chi-square variables

Sergey G. Bobkov, Alexey Naumov+1 more|arXiv (Cornell University)|Dec 19, 2020

Point processes and geometric inequalities8 references4 citations

TL;DR

This paper establishes sharp two-sided bounds for the maximum of the probability density function of a weighted sum of chi-square variables, both central and non-central. Using characteristic function analysis and inversion techniques, it proves that the sup-norm of the density is bounded above and below by constants times $(A_1A_2)^{-1/4}$, where $A_1 = \sum \lambda_k^2$ and $A_2 = \sum_{k\geq 2} \lambda_k^2$, with absolute constants explicitly quantified.

ABSTRACT

Two--sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.

Motivation & Objective

• To close the gap in the literature by providing lower bounds for the maximum density of a weighted sum of non-central chi-square variables.
• To establish dimension-free two-sided bounds for the sup-norm of the density function of a weighted sum of chi-square variables.
• To improve existing upper bounds in Gaussian comparison problems and high-dimensional central limit theorems by providing tight, uniform estimates.
• To analyze the behavior of the density maximum under non-central shifts, particularly when the largest eigenvalue does not dominate.

Proposed method

• Derive upper bounds using the Fourier inversion formula and characteristic function decay estimates.
• Establish lower bounds by constructing a set of positive measure where the density is bounded below using concentration and symmetry arguments.
• Apply Jensen’s inequality to the characteristic function to control the decay in the tail region.
• Use the monotonicity of logarithmic functions under constraints on eigenvalues to minimize the log-characteristic function.
• Employ a decomposition of the integral over the characteristic function into low- and high-frequency regions.
• Combine bounds from different frequency intervals to derive a uniform estimate for the sup-norm of the density function.

Experimental results

Research questions

• RQ1 What are sharp two-sided bounds for the maximum of the density function of a weighted sum of chi-square variables?
• RQ2 How do these bounds behave in the non-central case, particularly when the largest eigenvalue is not dominant?
• RQ3 Can the existing upper bounds for the density maximum be shown to be optimal up to absolute constants?
• RQ4 What is the role of the nuclear norm and eigenvalue structure in determining the sharpness of the bounds?
• RQ5 How do the bounds extend to the infinite-dimensional Hilbert space setting via finite-dimensional approximations?

Key findings

• For central chi-square sums, the maximum density satisfies $ c_0 (A_1 A_2)^{-1/4} \leq M(W_0) \leq c_1 (A_1 A_2)^{-1/4} $ with $ c_0 > 0.013 $, $ c_1 < 1.129 $, and $ A_1 = \sum \lambda_k^2 $, $ A_2 = \sum_{k\geq 2} \lambda_k^2 $.
• For non-central sums, under the condition $ \lambda_1^2 \leq A_1 / 3 $, the maximum density satisfies $ \frac{1}{4\sqrt{3}} (A_1 + B_1)^{-1/2} \leq M(W_a) \leq \frac{2}{\sqrt{A_1 + B_1}} $, where $ B_1 = \sum \lambda_k^2 a_k^2 $.
• The lower bound for $ M(W_a) $ holds without any restriction on $ \lambda_1^2 $, showing robustness of the estimate.
• The bounds are dimension-free and depend only on the eigenvalues and shift parameters, making them suitable for high- and infinite-dimensional applications.
• The results confirm the optimality of the upper bound in [1] by showing that the same order of magnitude is achieved from below.
• The proof technique provides a new, self-contained derivation of the upper bound in [1], independent of Kullback–Leibler divergence or Pinsker-type inequalities.