On the invariant measure of the random difference equation X_n=A_n X_n-1+ B_n in the critical case

2008/09/10 by Sara Brofferio, Brofferio, Sara, Dariusz Buraczewski +3
Mathematics · #60B15 #60G50 #60J10 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60B15 #msc:60G50 #msc:60J10

paper · pdf · doi:10.48550/arxiv.0809.1864

arxiv created 2008/11/10 · arxiv updated 2009/12/01

Abstract

We consider the autoregressive model on \R^d defined by the following stochastic recursion X_n = A_n X_n-1+B_n, where \(B_n,A_n)\ are i.i.d. random variables valued in \R^d× \R⁺. The critical case, when \E[log A₁]=0, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure ν for the Markov chain \X_n \. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on \R^d∖ \0\.

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