Limit theorems for some functionals with heavy tails of a discrete time Markov chain. (8th October 2014)

• Record Type: Journal Article
• Authors: Cattiaux, Patrick; Manou-Abi, Mawaki
• Is Part Of: ESAIM. Volume 18(2014)
• Page Start: 468
• Page End: 482
• Publication Date: 2014-10-08
• Subjects: Probabilities -- Periodicals; Mathematical statistics -- Periodicals; 519.2
• DOI: 10.1051/ps/2013043
• Languages: English
• ISSNs: 1292-8100

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (<italic>X</italic><sub><italic>n</italic></sub><italic>, n</italic> ≥ 0) with invariant distribution <italic>μ</italic>. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional <inline-formula><alternatives><tex-math id="tex_eq1"><![CDATA[\hbox{$S_{n}=\sum_{i=1}^{n}f(X_{i})$}]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgr3rd0" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq1" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="italic">S</mml:mi><mml:mi mathvariant="italic">n</mml:mi></mml:msub><mml:mo mathvariant="normal">=</mml:mo><mml:msubsup><mml:mo>âˆ'</mml:mo><mml:mrow><mml:mi mathvariant="italic">i</mml:mi><mml:mo mathvariant="normal">=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:mrow><mml:mi mathvariant="italic">n</mml:mi></mml:msubsup><mml:mi mathvariant="italic">f</mml:mi><mml:mo mathvariant="normal">(</mml:mo><mml:msub><mml:mi mathvariant="italic">X</mml:mi><mml:mi mathvariant="italic">i</mml:mi></mml:msub><mml:mo mathvariant="normal">)</mml:mo></mml:math></alternatives></inline-formula> for a possibly non square integrable function <italic>f</italic>. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence to stable distributions, obtained in [M. Denker and A. Jakubowski, <italic>Stat. Probab. Lett. </italic><bold>8 </bold>(1989) 477â€"483; M. Tyran-Kaminska, <italic>Stochastic Process. Appl. </italic><bold>120 </bold>(2010) 1629â€"1650; D. Krizmanic, Ph.D. thesis (2010); B. Basrak, D. Krizmanic and J. Segers, <italic>Ann. Probab. </italic><bold>40 </bold>(2012) 2008â€"2033] for stationary mixing sequences. Contrary to the usual <inline-formula><alternatives><tex-math id="tex_eq2"><![CDATA[L^2]]></tex-math><inline-graphic mimetype="image" xlink:href="ark:/27927/pgjcgr3r9f" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math id="mml_eq2" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo mathvariant="normal">L</mml:mo><mml:msubsup><mml:mo mathvariant="normal">2</mml:mo></mml:msubsup></mml:math></alternatives></inline-formula> framework studied in [P. Cattiaux, D. Chafai and A. Guillin, <italic>ALEA, Lat. Am. J. Probab. Math. Stat. </italic><bold>9 </bold>(2012) 337â€"382], where weak forms of ergodicity are sufficient to ensure the validity of the Central Limit Theorem, we will need here strong ergodic properties: the existence of a spectral gap, hyperboundedness (or hypercontractivity). These properties are also discussed. Finally we give explicit examples.</p> </abstract>