{"title":"On the affine recursion on R_d+ in the critical case","authors":"S. Brofferio, M. Peigné, Thi da Cam Pham","doi":"10.30757/ALEA.V18-37","DOIUrl":null,"url":null,"abstract":"We fix d ≥ 2 and denote S the semi-group of d× d matrices with non negative entries. We consider a sequence (An, Bn)n≥1 of i. i. d. random variables with values in S × R+ and study the asymptotic behavior of the Markov chain (Xn)n≥0 on R+ defined by: ∀n ≥ 0, Xn+1 = An+1Xn +Bn+1, where X0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices An equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (R+)d which is invariant for the chain (Xn)n≥0. The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices [16]. Its unicity is a consequence of a general property, called “local contractivity”, highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion [1] .","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"2012 1","pages":"1007"},"PeriodicalIF":0.6000,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/ALEA.V18-37","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 2
Abstract
We fix d ≥ 2 and denote S the semi-group of d× d matrices with non negative entries. We consider a sequence (An, Bn)n≥1 of i. i. d. random variables with values in S × R+ and study the asymptotic behavior of the Markov chain (Xn)n≥0 on R+ defined by: ∀n ≥ 0, Xn+1 = An+1Xn +Bn+1, where X0 is a fixed random variable. We assume that the Lyapunov exponent of the matrices An equals 0 and prove, under quite general hypotheses, that there exists a unique (infinite) Radon measure λ on (R+)d which is invariant for the chain (Xn)n≥0. The existence of λ relies on a recent work by T.D.C. Pham about fluctuations of the norm of product of random matrices [16]. Its unicity is a consequence of a general property, called “local contractivity”, highlighted about 20 years ago by M. Babillot, Ph. Bougerol et L. Elie in the case of the one dimensional affine recursion [1] .
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.