On some explicit limiting distributions related to free multiplicative law of large numbers

Abstract

Suppose $\mathbb {V}_{\nu }(\cdot )$ is the pseudo-variance function of the Cauchy–Stieltjes Kernel (CSK) family ${\mathcal {K}_{-}}(\nu )$ generated by a non degenerate probability measure $\nu$ on the positive real line. Denote by $\Phi (\nu )$ the law of large numbers for free multiplicative convolution given in [17]. An explicit expression of $\Phi (\nu )$ is given in [14] in terms of the pseudo-variance function $\mathbb {V}_{\nu }(.)$. In this paper, we give explicitly in terms of the pseudo-variance function $\mathbb {V}_{\nu }(.)$ the limiting distributions $\Phi (\nu ^{\boxtimes t})$, $\Phi ((D_{1/s}(\nu ^{\boxplus s}))^{\boxtimes t})$, $\Phi ((D_{1/r}(\nu ^{\uplus r}))^{\boxtimes t})$ and $\Phi ((\mathbb {B}_r(\nu ))^{\boxtimes t})$ for $s>1$, $t>1$ and $r>0$, where $\mathbb {B}_r(\nu )=(\nu ^{\boxplus 1+r})^{\uplus \frac{1}{1+r}}$ and $D_c(\nu )$ denotes the dilation of measure $\nu$ by a number $c\ne 0$. Some examples of calculations of $\Phi (\nu ^{\boxtimes t})$, $\Phi ((D_{1/s}(\nu ^{\boxplus s}))^{\boxtimes t})$, $\Phi ((D_{1/r}(\nu ^{\uplus r}))^{\boxtimes t})$ and $\Phi ((\mathbb {B}_r(\nu ))^{\boxtimes t})$ are given for probability measures $\nu$ of importance in noncommutative probability.