On infinitely divisible distributions with polynomially decaying characteristic functions

TL;DR

This paper characterizes when infinitely divisible distributions have characteristic functions that decay polynomially, linking this decay to Fourier multipliers on Besov spaces under mild regularity conditions.

Contribution

It establishes necessary and sufficient conditions for polynomial decay of characteristic functions of infinitely divisible distributions and relates this decay to Fourier multipliers on Besov spaces.

Findings

Polynomial decay of characteristic functions characterized

Equivalence between decay and Fourier multiplier property

Conditions applicable under mild regularity assumptions

Abstract

We provide necessary and sufficient conditions on the characteristics of an infinitely divisible distribution under which its characteristic function $\phi$ decays polynomially. Under a mild regularity condition this polynomial decay is equivalent to $1/\phi$ being a Fourier multiplier on Besov spaces.