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 $\int_0^{\infty-} f(s) dX_s^{(\rho)}$, where $f(s)$ is a non-random function and $\{X_s^{(\rho)}\}$ is a L\'evy 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.