Local limit theorems on some non unimodular groups

Abstract

Let $G_d$ be the semi-direct product of ${\mathbb{R}}^{\ast +}$ and ${\mathbb{R}}^d$, $d\ge 1$ and let us consider the product group $G_{d,N}=G_d\times {\mathbb{R}}^N$, $N\ge 1$. For a large class of probability measures $\mu$ on $G_{d,N}$, one proves that there exists $\rho (\mu )\in [0,1]$ such that the sequence of finite measures

converges weakly to a nondegenerate measure.