Tail-homogeneity of stationary measures for some multidimensional stochastic recursions

We consider a stochastic recursion X n +1  =  M n +1 X n  +  Q n +1 , ( ), 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 and . In the present paper we show that if the recursion has a unique stationary measure ν with unbou...

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Bibliographic Details
Published in:Probability theory and related fields 2009-11, Vol.145 (3-4), p.385-420, Article 385
Main Authors: Buraczewski, Dariusz, Damek, Ewa, Guivarc’h, Yves, Hulanicki, Andrzej, Urban, Roman
Format: Article
Language:English
Subjects:
Economics
Euclidean geometry
Euclidean space
Exact sciences and technology
Finance
General topics
Harmonic analysis
Homogeneity
Insurance
Management
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Probability
Probability and statistics
Probability Theory and Stochastic Processes
Quantitative Finance
Random variables
Sciences and techniques of general use
Statistics for Business
Theoretical
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Summary:We consider a stochastic recursion X n +1  =  M n +1 X n  +  Q n +1 , ( ), 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 and . 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.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-008-0172-8