A simple variance inequality for -statistics of a Markov chain with applications

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• 作者：G. Fort^a ; ^{gersende.fort@telecom-paristech.fr} ; ^{gfort@tsi.enst.fr} ; E. Moulines^a ; ^{eric.moulines@telecom-paristech.fr} ; P. Priouret^b ; ^{p.priouret@upmc.jussieu.fr} ; P. Vandekerkhove^c ; ^{Pierre.Vandekerkhove@univ-mlv.fr}
• 关键词：U-statistics ; Markov chains ; Inequalities ; Limit theorems ; Law of large numbers
• 刊名：Statistics and Probability Letters
• 出版年：2012
• 期刊代码：56_01677152
• 类别：mt
• 出版时间：June, 2012
• 卷：82
• 期：6
• 页码：1193-1201
• 文件大小：259 K

摘要

We establish a simple variance inequality for -statistics whose underlying sequence of random variables is an ergodic Markov Chain. The constants in this inequality are explicit and depend on computable bounds on the mixing rate of the Markov Chain. We apply this result to derive the strong law of large numbers for -statistics of a Markov Chain under conditions which are close to being optimal.