On the invariant measure of the random difference equation $X_n=A_n X_{n-1}+ B_n$ in the critical case

By S. Brofferio, Dariusz Buraczewski, Ewa Damek. 2008

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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 $\R^d\times \R^+$. The critical case, when $\E\big[\log A_1\big]=0$, was studied by Babillot, Bougeorol and Elie, who proved that there exists a unique invariant Radon measure $\nu$ for the Markov chain $\{X_n \}$. In the present paper we prove that the weak limit of properly dilated measure $\nu$ exists and defines a homogeneous measure on $\R^d\setminus \{0\}$.

Subfields: Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5e1a214f732f9b05573ea42962fb7d6a1ceb2fbd