Spectral gap properties for linear random walks and Pareto's asymptotics for affine stochastic recursions (Q297434)

In this note, the authors investigate the asymptotic properties of the iterated convolutions \(\mu^n\ast\delta_v\), (resp. \(\lambda^n\ast\delta_v\)), where \(\mu\) is a probability measure on the linear group \(G=\mathrm{GL}(V)\), \(V=\mathbb R^d\) (\(d\)-dimensional Euclidean space); here \(\lambda\) is a probability measure on \(H=\mathrm{Aff}(V)\) with projection \(\mu\) on \(G\) such that \(\operatorname{supp}\mu\) has no fixed point in \(V\).NEWLINENEWLINEThe sequence \(\lambda^n\ast\delta_v\) converges weakly to a probability measure \(\rho\). One of the main results of this note gives a so-called homogeneity of \(\rho\) at infinity, i.e., Pareto's asymptotics of \(\rho\).