Convergence to stable laws for a class of multidimensional stochastic recursions

• Published in: Probability theory and related fields 2010-11, Vol.148 (3-4), p.333-402
• Main Authors: Buraczewski, Dariusz, Damek, Ewa, Guivarc’h, Yves
• Format: Article
• Language: English
• Subjects: Affine transformations; Analogies; Characteristic functions; Convergence; Economics; Error analysis; Errors; Exact sciences and technology; Finance; General topics; Homogeneity; Hyperplanes; Hypotheses; Insurance; Limit theorems; Management; Markov analysis; Markov chains; Markov processes; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Perturbation; Phase transitions; Probability; Probability and statistics; Probability Theory and Stochastic Processes; Probability theory on algebraic and topological structures; Quantitative Finance; Random variables; Recursion; Sciences and techniques of general use; Spectrum analysis; Statistics for Business; Stochasticity; Sums; Syntax; Theoretical; Thermal expansion
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-009-0233-7

We consider a Markov chain on defined by the stochastic recursion X n  =  M n X n -1  +  Q n , where ( Q n , M n ) are i.i.d. random variables taking values in the affine group . Assume that M n takes values in the group of similarities of , and the Markov chain has a unique stationary measure ν, which has unbounded support. We denote by | M n | the expansion coefficient of M n and we assume for some positive α . We show that the partial sums , properly normalized, converge to a normal law ( α ≥ 2) or to an infinitely divisible law, which is stable in a natural sense ( α   2) or of the tails of ν and of stationary measure for an associated Markov operator ( α ≤ 2). We extend the results to the situation where M n is a random generalized similarity.