Comparing moments of real log-concave random variables

2022/11/09 by Murawski, Daniel
#60E15 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2211.05210

Abstract

We show that for every mean zero log-concave real random variable X one has ‖X‖_p ≤ (p)/(q) ‖X‖_q for p ≥ q ≥ 1, going beyond the well-known case of symmetric random variables. We also prove that in the class of arbitrary log-concave real random variables for p>q > 0 the quantity ‖X‖_p / ‖X‖_q is maximized for some shifted exponential distribution. Building upon this we derive the bound ‖X‖_p ≤ C₀ (p)/(q) ‖X‖_q for arbitrary log-concave X, with best possible absolute constant C₀=e^W(1/e) ≈ 1.3211 in front of (p)/(q), where W stands for the Lambert function.

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