Weak drifts of infinitely divisible distributions and their applications

TL;DR

This paper introduces the concept of weak drift for infinitely divisible distributions, explores its properties, and applies it to characterize limits, laws of large numbers, and behavior of Lévy processes near zero.

Contribution

It defines weak drift for infinitely divisible distributions, relates it to weak mean via inversion, and applies these concepts to stochastic integral mappings and Lévy process behaviors.

Findings

Weak drift equals minus weak mean of the inversion when no Gaussian part.

Characterization of ranges and limits of iterative stochastic integral mappings.

Necessary and sufficient conditions for weak law of large numbers and behavior near zero for Lévy processes.

Abstract

Weak drift of an infinitely divisible distribution $\mu$ on $\mathbb{R}^d$ is defined by analogy with weak mean; properties and applications of weak drift are given. When $\mu$ has no Gaussian part, the weak drift of $\mu$ equals the minus of the weak mean of the inversion $\mu'$ of $\mu$. Applying the concepts of having weak drift 0 and of having weak drift 0 absolutely, the ranges, the absolute ranges, and the limit of the ranges of iterations are described for some stochastic integral mappings. For L\'{e}vy processes the concepts of weak mean and weak drift are helpful in giving necessary and sufficient conditions for the weak law of large numbers and for the weak version of Shtatland's theorem on the behavior near $t=0$; those conditions are obtained from each other through inversion.