Moments of sums of exponentials, beyond CHS

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.

💡 Research Summary

The paper investigates two fundamental problems concerning linear combinations of independent standard exponential random variables (E_1,E_2,\dots). The first problem is to determine the sharp lower bound for the (L_p)-norm of such a sum when the variance is fixed, for all (p\ge 2). The authors prove (Theorem 1) that for any real coefficients (x_1,\dots,x_n) and (X=\sum_{j=1}^n x_jE_j), \