Risk Bounds for Infinitely Divisible Distribution

TL;DR

This paper derives new risk bounds for samples from infinitely divisible distributions using martingale methods, providing faster convergence rates than traditional empirical process bounds.

Contribution

It introduces two deviation inequalities for ID distributions with zero Gaussian component and analyzes their implications for risk bounds and convergence rates.

Findings

Developed two deviation inequalities for ID distributions

Established risk bounds based on covering numbers

Showed faster asymptotic convergence rates than generic i.i.d. bounds

Abstract

In this paper, we study the risk bounds for samples independently drawn from an infinitely divisible (ID) distribution. In particular, based on a martingale method, we develop two deviation inequalities for a sequence of random variables of an ID distribution with zero Gaussian component. By applying the deviation inequalities, we obtain the risk bounds based on the covering number for the ID distribution. Finally, we analyze the asymptotic convergence of the risk bound derived from one of the two deviation inequalities and show that the convergence rate of the bound is faster than the result for the generic i.i.d. empirical process (Mendelson, 2003).