Moments of sums of exponentials, beyond CHS

TL;DR

This paper derives a precise lower bound on the $L_p$-norm of sums of independent exponential variables for $p \,\geq\, 2$, extending classical results and analyzing Schur-monotonicity regimes.

Contribution

It introduces a sharp lower bound on the $L_p$-norm for sums of exponentials and characterizes the $p$-regimes where Schur-monotonicity holds, extending Hunter's theorem.

Findings

Established a sharp lower bound on $L_p$-norms for exponential sums.

Identified the $p$-regime where Schur-monotonicity applies.

Extended Hunter's positivity theorem to a broader setting.

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.