Some notes on moment inequalities for heavy-tailed distributions

TL;DR

This paper explores the relationship between moments and tail behaviors of heavy-tailed distributions, providing sharp inequalities, examples, and concentration bounds for polynomial chaos of any order.

Contribution

It introduces new moment inequalities for heavy-tailed distributions and derives concentration bounds for polynomial chaos, enhancing understanding of tail-moment relations.

Findings

Established sharp moment inequalities for Pareto-type distributions

Derived concentration bounds for polynomial chaos of any order

Provided examples demonstrating the sharpness of results

Abstract

We investigate the relation between moments and tails of heavy-tailed (in particular, Pareto-type) distributions. We also discuss the sharpness of our results in a number of examples under certain regularity conditions like log-convexity. Moreover, we derive concentration bounds for polynomial chaos of any order $d$.