On the invariant measure of the random difference equation \(X_{n} = a_{n}x_{n - 1} + b_{n}\) in the critical case
random coefficients autoregressive model
Let \(G\) be the affine group \(\mathbb{R}^d\times\mathbb{R}^+\) with the multiplication \((b,a)\cdot(b', a')= (b+ ab, aa)\). \(G\) acts on \(x\in\mathbb{R}^d\) by \((b, a)\cdot x= ax+ b\), \((b,a)\in G\). The authors consider the autoregressive process on \(\mathbb{R}^d\), i.e., NEWLINE\[NEWLINEX^x_0= x,\;X^x_n= A_n X^x_{n-1}+ B_n,\quad n= 1,2,\dotsc,NEWLINE\]NEWLINE where the random pairs \((B_n,A_n)\) are independent, identically distributed with a common distribution \(\mu\). In group terms, the Markov chain \(X^x_n\) may be written NEWLINE\[NEWLINEX^x_n= (B_n, A_n)\cdot(B_{n-1}, A_{n-1})\dotsm(B_1, A_1)\cdot x,NEWLINE\]NEWLINE i.e., the action on \(x\) of a random walk on \(G\).NEWLINENEWLINE Here, the critical case \(\operatorname{E}[\log A_1]= 0\) is considered. The Markov chain \(X^x_n\) admits a unique (up to a multiplicative constant) invariant Radon measure \(\nu\). Refining and generalizing the work by \textit{M. Babillot}, \textit{P. Bougerol} and \textit{L. Elie} [Ann. Probab. 25, No. 1, 478--493 (1997; Zbl 0873.60045)], the authors are able to describe the asymptotics of \(\nu\) quite explicitly. Under some natural non-periodicity and non-degeneracy conditions and some additional moment requirements they prove, e.g., that, for all \(0<\alpha<\beta\), NEWLINE\[NEWLINE\lim_{z\to\infty}\,\nu\{u:\alpha z<|u|<\beta z\}= C_+\log{\beta\over\alpha}NEWLINE\]NEWLINE for some positive constant \(C_+\). Results are also obtained in the case when \(A_1\) is periodic (the distribution of \(A_1\) is supported on a multiplicative lattice).
scientific article; zbMATH DE number 6042932
scientific article