General upper and lower tail estimates using Malliavin calculus and Stein's equations

TL;DR

This paper develops a general technique combining Malliavin calculus and Stein's equations to compare the tails of random variables to various reference distributions, extending previous Gaussian-focused methods.

Contribution

It introduces a versatile approach for tail comparison using Malliavin calculus and Stein's equations, applicable to a broad class of distributions beyond Gaussian.

Findings

Provides concrete tail bounds for power and exponential distributions.

Extends the Nourdin-Peccati strategy to Pearson distributions.

Offers detailed analysis of Stein equation solutions for tail estimates.

Abstract

Following a strategy recently developed by Ivan Nourdin and Giovanni Peccati, we provide a general technique to compare the tail of a given random variable to that of a reference distribution. This enables us to give concrete conditions to ensure upper and/or lower bounds on the random variable's tail of various power or exponential types. The Nourdin-Peccati strategy analyzes the relation between Stein's method and the Malliavin calculus, and is adapted to dealing with comparisons to the Gaussian law. By studying the behavior of the solution to general Stein equations in detail, we show that the strategy can be extended to comparisons to a wide class of laws, including many Pearson distributions.