Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

• 成果类型: Article
• 署名作者: Mirek, Mariusz
• 署名单位: University of Wroclaw
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-010-0312-9
• 发表日期: 2011
• 页码: 705-734
• 关键词: CENTRAL LIMIT-THEOREMS; additive-functionals; renewal theory; markov-chain

摘要

We consider the Markov chain {X-n(x)}(n=0)(infinity) = 0 on R-d defined by the stochastic recursion X-n(x) = psi(theta n) (X-n-1(x)), starting at x is an element of R-d, where theta(1), theta(2), ... are i.i.d. random variables taking their values in a metric space (Theta, r), and psi(theta n) : R-d -> R-d are Lipschitz maps. Assume that theMarkov chain has a unique stationary measure.. Under appropriate assumptions on psi(theta n) , we will show that the measure. has a heavy tail with the exponent alpha > 0 i. e..({x. R-d : |x| > t}) (sic) t(-alpha). Using this result we show that properly normalized Birkhoff sums S-n(x) = Sigma(n)(k=1) X-k(x), converge in law to an alpha-stable law for alpha is an element of (0, 2].