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On invariant measures of stochastic recursions in a critical case. (English) Zbl 1151.60034
The present paper is devoted to the study of the autoregressive model on \(\mathbb{R}\) described by the following recurrence equation:
\[ X_n=A_nX_{n-1}+B_n, \] where \((B_n,A_n)\in \mathbb{R}\times \mathbb{R}^+\) are i.i.d. random variables according to a probability measure \(\mu\) and \(A_1\) satisfies the following condition: \(\mathbb{E}[A_1]=0\) (critical case). From another point of view, we may think of \((B_n,A_n)\) as a random element in the “\(ax+b\)” group and consider \(A_nX_{n-1}+B_n\) as the \((B_n,A_n)\)-image of \(X_{n-1}\). In [Ann. Probab. 25, No. 1, 478–493 (1997; Zbl 0873.60045)], M. Babillot, P. Bougerol and L. Elie showed that there exists a unique (up to a constant) invariant Radon measure \(\nu\) for the process \(\{X_n\}\) and that the following asymptotic estimate holds: for all positive \(\alpha,\beta\) one has \[ \nu((\alpha x,\beta x])\sim \log(\beta/\alpha)\cdot L^{\pm}(| x| ) \] as \(x\rightarrow \pm \infty\), where \(L^\pm\) are slowly varying functions.
The main result in the paper under review is the following: if \(\mu\) is spread-out and has some moments, then the slowly varying functions are constant. To reach his goal, the author also derives results on a related Poisson equation and on the asymptotic behavior of its solutions. In the final section of the paper, the author consider two other random models. The first is of the form \(X'_n=B_n+A_n \max\{X_{n-1}',C_n\}\), and was introduced by [G. Letac, A contraction principle for certain Markov chains and its applications. Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 263–273 (1986; Zbl 0587.60057)]. The second is of the form \(X_n''=\max\{A_n,X''_{n-1},D_n\}\) and is widely studied in models for the waiting time in a single server queue.

MSC:
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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