Abstract
We consider a stochastic recursion X n+1 = M n+1 X n + Q n+1, (\({n\in \mathbb {N}}\)), where (Q n , M n ) are i.i.d. random variables such that Q n are translations, M n are similarities of the Euclidean space \({\mathbb {R}^d}\) and \({X_n\in \mathbb {R}^d}\). In the present paper we show that if the recursion has a unique stationary measure ν with unbounded support then the weak limit of properly dilated ν exists and defines a homogeneous tail measure Λ. The structure of Λ is studied and the supports of ν and Λ are compared. In particular, we obtain a product formula for Λ.
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This research project has been partially supported by European Commission via IHP Network 2002–2006 Harmonic Analysis and Related Problems(contract Number: HPRN-CT-2001-00273-HARP) and a Marie Curie Transfer of Knowledge Fellowship Harmonic Analysis, Nonlinear Analysis and Probability (contract number MTKD-CT-2004-013389). D. Buraczewski, E. Damek, A. Hulanicki and R. Urban were also supported by KBN grants 1 P03A 018 26 and N201 012 31/1020.
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Buraczewski, D., Damek, E., Guivarc’h, Y. et al. Tail-homogeneity of stationary measures for some multidimensional stochastic recursions. Probab. Theory Relat. Fields 145, 385–420 (2009). https://doi.org/10.1007/s00440-008-0172-8
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DOI: https://doi.org/10.1007/s00440-008-0172-8
Mathematics Subject Classification (2000)
- 60J10
- 60K05