Spectral gap properties and convergence to stable laws for affine random walks on $\mathbb{R}^d$
Résumé
We consider a Markov chain $\{X_n\}_{n=1}^{\infty}$ on $\mathbb{R}^d$ defined by the stochastic recursion $X_n =M_n X_{n-1}+Q_n$ where $(Q_n,M_n)$ are i.i.d. random variables which take values in the affine group of the vector space $\mathbb{R}^d$. Under natural hypothesis on $(Q_n,M_n)$, including negativity of the dominant Lyapunov exponent of the product of the matrices $M_n$, the transition operator $P$ of the chain has a unique stationary measure $\eta$ and it is known that $\eta$ is $\alpha$-homogeneous at infinity for some $\alpha>0$ depending on the law of $M_n$. We show spectral gap properties for $P$ and for a class of Fourier operators $P_v (v\in \mathbb{R}^d)$, on function spaces on $\mathbb{R}^d$ of H\"{o}lder type. If $d>1$, we consider the Birkhoff sums $S_n= \sum_{k=0}^n X_k$ and we show that the normalized sums converge to a normal law ($\alpha\geq 2$) or to an $\alpha$-stable law $(\alpha<2)$ on $ \mathbb{R}^d$, and these laws are fully non-degenerate. For $d=1$ such results were first obtained in Guivarc'h and Le Page(2008). Here we describe their natural extension to the general multidimensional setting. The corresponding analysis of the characteristic function of $S_n$ is based on the spectral gap properties of $P_v$ and on the homogeneity at infinity of $\eta$ and of a family of companion measures $\eta_v$.