Description of limits of ranges of iterations of stochastic integral mappings of infinitely divisible distributions

Abstract

For infinitely divisible distributions $\rho$ on ${\mathbb{R}}^d$ the stochastic integral mapping ${\Phi }_f\rho$ is defined as the distribution of improper stochastic integral ${\int }_0^{\infty -}f(s)d{X}_s^{(\rho )}$, where $f(s)$ is a non-random function and $\{X_s^{(\rho )}\}$ is a Lévy process on ${\mathbb{R}}^d$ with distribution $\rho$ at time 1. For three families of functions $f$ with parameters, the limits of the nested sequences of the ranges of the iterations ${\Phi }_f^n$ are shown to be some subclasses, with explicit description, of the class $L_{\infty }$ of completely selfdecomposable distributions. In the critical case of parameter 1, the notion of weak mean 0 plays an important role. Examples of $f$ with different limits of the ranges of ${\Phi }_f^n$ are also given.

16 pages. To appear in ALEA Lat. Am. J. Probab. Math. Statist

References in corpus (2)

• Transformations of infinitely divisible distributions via improper stochastic integrals
• Type A Distributions: Infinitely Divisible Distributions Related to Arcsine Density