Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Limit theory for some positive stationary processes with infinite mean

Jon Aaronson and Roland Zweimüller

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Abstract

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.

Résumé

Nous prouvons des théorèmes limites et des lois du logarithme itéré unilatérales pour une classe de processus stochastiques positifs, mélangeants et stationnaires. Cette classe contient en particulier les processus obtenus par des observables nonintégrables de certaines applications dilatantes. Ceci est obtenu en généralisant la théorie de Darling–Kac à une famille appropriée de transformations préservant la mesure.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 256-284.

Dates
First available in Project Euclid: 1 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1388545274

Digital Object Identifier
doi:10.1214/12-AIHP513

Mathematical Reviews number (MathSciNet)
MR3161531

Zentralblatt MATH identifier
1291.60067

Subjects
Primary: 60Fxx: Limit theorems [See also 28Dxx, 60B12]
Secondary: 37A40: Nonsingular (and infinite-measure preserving) transformations 60G10: Stationary processes

Keywords
Infinite invariant measure Transfer operator Infinite ergodic theory Darling–Kac theorem Pointwise dual ergodic Mixing coefficient Stable limit One-sided law of iterated logarithm

Citation

Aaronson, Jon; Zweimüller, Roland. Limit theory for some positive stationary processes with infinite mean. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 256--284. doi:10.1214/12-AIHP513. https://projecteuclid.org/euclid.aihp/1388545274


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