2020/03/20 by Sara Brofferio, Marc Peigné, Brofferio, Sara +3
Mathematics · #Stochastic processes and statistical mechanics #Mathematical Dynamics and Fractals #Markov Chains and Monte Carlo Methods
paper · pdf · doi:10.48550/arxiv.2003.09380
We fix d ≥ 2 and denote \mathcal S the semi-group of d × d matrices with non negative entries. We consider a sequence (An, Bn)n ≥ 1 of i. i. d. random variables with values in \mathcal S× \mathbb R+d and study the asymptotic behavior of the Markov chain (Xn)n ≥ 0 on \mathbb R+d defined by: ∀ n ≥ 0, Xn+1=An+1Xn+Bn+1, where X0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices An equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (\mathbb R+)d which is invariant for the chain (Xn)n ≥ 0. The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices . Its unicity is a consequence of a general property, called "local contractivity", highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion .