Entropies of sums of independent gamma random variables

TL;DR

This work studies weighted sums $X_a=\sum_{j=1}^n a_jX_j$ of i.i.d. gamma variables $X_j\sim\Gamma(\gamma)$, aiming to quantify Gaussianity via relative entropy and Rényi entropies. It introduces a mgf-based approach to prove Schur-convexity/concavity in the weight vector under centering and without, extracting entropy bounds and moment- and density-based extremal results. The main contributions include (i) simple proofs of Schur-type results that recover and extend BNZ's gamma-extremality, (ii) nonasymptotic Rényi-entropy bounds for $h_α$ with $α>1$ and varying $\gamma,n$, and (iii) generalized maximum-density bounds for sums with $\Gamma(\gamma)$ components (extending BNU) with explicit regime-dependent constants. These results have implications for anti-concentration and Gaussian quadratic forms, and they provide streamlined techniques for analyzing entropy extremality in weighted gamma-sums.

Abstract

We establish several Schur-convexity type results under fixed variance for weighted sums of independent gamma random variables and obtain nonasymptotic bounds on their Rényi entropies. In particular, this pertains to the recent results by Bartczak-Nayar-Zwara as well as Bobkov-Naumov-Ulyanov, offering simple proofs of the former and extending the latter.

Theorems & Definitions (27)

• Theorem 1
• Theorem 2
• Corollary 3: Bartczak-Nayar-Zwara, BNZ
• Corollary 4
• Theorem 5
• Theorem 6
• Remark 7
• Lemma 8
• proof
• proof : Proof of Theorem \ref{['thm:Phi']}