Moments of sums of exponentials, beyond CHS
Silouanos Brazitikos, Colin Tang, Tomasz Tkocz
TL;DR
The paper addresses sharp forward $L_p$ bounds for sums $X=\sum_{j=1}^n x_j\mathscr{E}_j$ of independent exponential random variables, establishing $\|X\|_p \ge \|G\|_p\sqrt{\mathrm{Var}(X)}$ for all $p\ge 2$, where $G\sim N(0,1)$, thus extending Hunter's positivity theorem for CHS polynomials to all even and non-even moments in this setting. It then identifies the exact regimes where the map $x\mapsto \mathbb{E}|\sum_j x_j\mathscr{E}_j|^p$ is Schur-monotone for low moments, using integral representations and monotonicity arguments; notably, Schur-monotonicity holds for $-1<p<4$ but fails for $p>4$. The proofs combine a two-step strategy adapting Hunter's approach—a local critical-point analysis to reduce to $2\le p\le 4$ and a Fourier-analytic bound to extend to all $p\ge 2$—with integral-moment representations that yield sharp comparisons to Gaussian moments. The work further offers a gamma-distribution generalization and discusses log-convexity properties, suggesting the Gaussian extremiser persists in broader families and linking the probabilistic problem to simplex-geometry questions via moment methods for CHS polynomials.
Abstract
We establish a sharp lower bound on the $L_p$-norm of sums of independent exponential random variables with fixed variance, for $p \geq 2$, thus extending Hunter's positivity theorem (1976) for completely homogeneous polynomials. We determine the exact regime of $p$ where such sums enjoy Schur-monotonicity.