Weak drifts of infinitely divisible distributions and their applications

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.