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Heavy tail properties of stationary solutions of multidimensional stochastic recursions. (English) Zbl 1126.60052
Denteneer, Dee (ed.) et al., Dynamics and stochastics. Festschrift in honor of M. S. Keane. Selected papers based on the presentations at the conference ‘Dynamical systems, probability theory, and statistical mechanics’, Eindhoven, The Netherlands, January 3–7, 2005, on the occasion of the 65th birthday of Mike S. Keane. Beachwood, OH: IMS, Institute of Mathematical Statistics (ISBN 0-940600-64-1/pbk). Institute of Mathematical Statistics Lecture Notes - Monograph Series 48, 85-99 (2006).
Summary: We consider the following recurrence relation with random i.i.d. coefficients \((a_n,b_n)\): \[ x_{n+1}=a_{n+1} x_n+b_{n+1} \] where \(a_n\in \text{GL}(d,\mathbb R),b_n\in \mathbb R^d\). Under natural conditions on \((a_n,b_n)\) this equation has a unique stationary solution, and its support is non-compact. We show that, in general, its law has a heavy tail behavior and we study the corresponding directions. This provides a natural construction of laws with heavy tails in great generality. Our main result extends to the general case, the results previously obtained by H. Kesten in [Acta Math. 131, 207–248 (1973; Zbl 0291.60029)] under positivity or density assumptions, and the results recently developed in [C. Kl├╝ppelberg and S. Pergamenchtchikov, Ann. Appl. Probab. 14, No. 2, 971–1005 (2004; Zbl 1094.62114] in a special framework.
For the entire collection see [Zbl 1113.60008].

MSC:
60H25 Random operators and equations (aspects of stochastic analysis)
60G50 Sums of independent random variables; random walks
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
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