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

For infinitely divisible distributions $\rho$ on ${\mathbb{R}}^d$ the stochastic integral mapping ${\mathrm{\Phi }}_f\rho$ is defined as the distribution of improper stochastic integral ${\int }_0^{\mathrm{\infty }-}f(s)d{X}_s^{(\rho )}$, 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 ${\mathrm{\Phi }}_f^n$ are shown to be some subclasses, with explicit description, of the class $L_{\mathrm{\infty }}$ 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 ${\mathrm{\Phi }}_f^n$ are also given.