Local limit theorems on some non unimodular groups (Q1283366)

Let \(G_d\) be the semi-direct product of \(R^{*+}\) and \(R^d, d\geq 1\), and let us consider the product group \(G_{d,N}=G_d\times R^N, N\geq 1\). 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 \(\{\frac{n^{(N+3)/2}}{\rho (\mu)^n}\mu^{*n}\}_{n\geq 1}\) converges weakly to a non-degenerate measure.