Local limit theorems on some non unimodular groups

Let Gd be the semi-direct product of R*(+) and R-d, d greater than or equal to 1 and let us consider the product group G(d,N) = G(d) x R-N, N greater than or equal to 1 For a large class of probability measures mu on G(d,N), One proves that there exists rho(mu) epsilon]0, 1] such that the sequence of finite {n((N+3)/2)/rho(mu)(n) mu*(n)}(n greater than or equal to 1) measures converges weakly to a non-degenerate measure.