Limit theory for some positive stationary processes with infinite mean
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 256-284.

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.

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.

DOI: 10.1214/12-AIHP513
Classification: 60Fxx, 37A40, 60G10
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
@article{AIHPB_2014__50_1_256_0,
     author = {Aaronson, Jon and Zweim\"uller, Roland},
     title = {Limit theory for some positive stationary processes with infinite mean},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {256--284},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {1},
     year = {2014},
     doi = {10.1214/12-AIHP513},
     mrnumber = {3161531},
     zbl = {1291.60067},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/12-AIHP513/}
}
TY  - JOUR
AU  - Aaronson, Jon
AU  - Zweimüller, Roland
TI  - Limit theory for some positive stationary processes with infinite mean
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2014
SP  - 256
EP  - 284
VL  - 50
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/12-AIHP513/
DO  - 10.1214/12-AIHP513
LA  - en
ID  - AIHPB_2014__50_1_256_0
ER  - 
%0 Journal Article
%A Aaronson, Jon
%A Zweimüller, Roland
%T Limit theory for some positive stationary processes with infinite mean
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2014
%P 256-284
%V 50
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/12-AIHP513/
%R 10.1214/12-AIHP513
%G en
%F AIHPB_2014__50_1_256_0
Aaronson, Jon; Zweimüller, Roland. Limit theory for some positive stationary processes with infinite mean. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 256-284. doi : 10.1214/12-AIHP513. http://archive.numdam.org/articles/10.1214/12-AIHP513/

[1] J. Aaronson. An Introduction to Infinite Ergodic Theory. Am. Math. Soc., Providence, RI, 1997. | MR | Zbl

[2] J. Aaronson. The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39 (1981) 203-234. | MR | Zbl

[3] J. Aaronson. Random f-expansions. Ann. Probab. 14 (1986) 1037-1057. | MR | Zbl

[4] J. Aaronson and M. Denker. Lower bounds for partial sums of certain positive stationary processes. In Almost Everywhere Convergence (Columbus, OH, 1988) 1-9. Academic Press, Boston, MA, 1989. | MR | Zbl

[5] J. Aaronson and M. Denker. Upper bounds for ergodic sums of infinite measure preserving transformations. Trans. Amer. Math. Soc. 319 (1990) 101-138. | MR | Zbl

[6] J. Aaronson and M. Denker. On the FLIL for certain ψ-mixing processes and infinite measure preserving transformations. C. R. Acad. Sci. Paris Sèr. I Math. 313 (1991) 471-475. | MR | Zbl

[7] J. Aaronson, M. Denker, O. Sarig and R. Zweimüller. Aperiodicity of cocycles and conditional local limit theorems. Stoch. Dyn. 4 (2004) 31-62. | MR | Zbl

[8] J. Aaronson and H. Nakada. On the mixing coefficients of piecewise monotonic maps. Israel J. Math. 148 (2005) 1-10. | MR | Zbl

[9] N. H. Bingham. Limit theorems for occupation times of Markov processes. Z. Wahrsch. verw. Gebiete 17 (1971) 1-22. | MR | Zbl

[10] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge, 1989. | MR | Zbl

[11] R. C. Bradley Jr. On the ψ-mixing condition for stationary random sequences. Trans. Amer. Math. Soc. 276 (1983) 55-66. | MR | Zbl

[12] R. C. Bradley Jr. Introduction to Strong Mixing Conditions 1. Kendrick Press, Heber City, UT, 2007. | Zbl

[13] R. A. Davis. Stable limits for partial sums of dependent random variables. Ann. Probab. 11 (1983) 262-269. | MR | Zbl

[14] D. A. Darling and M. Kac. On occupation times for Markoff processes. Trans. Amer. Math. Soc. 84 (1957) 444-458. | MR | Zbl

[15] B. I. Korenblyum. On the asymptotic behavior of Laplace integrals near the boundary of a region of convergence. Dokl. Akad. Nauk SSSR (N.S.) 104 (1955) 173-176. | MR

[16] M. Rychlik. Bounded variation and invariant measures. Studia Math. 76 (1983) 69-80. | MR | Zbl

[17] J. D. Samur. Convergence of sums of mixing triangular arrays of random vectors with stationary rows. Ann. Probab. 12 (1984) 390-426. | MR | Zbl

[18] M. Thaler. Transformations on [0,1] with infinite invariant measures. Israel J. Math. 46 (1983) 67-96. | MR | Zbl

[19] M. Thaler. The Dynkin-Lamperti arc-sine laws for measure preserving transformations. Trans. Amer. Math. Soc. 350 (1998) 4593-4607. | MR | Zbl

[20] M. Thaler and R. Zweimüller. Distributional limit theorems in infinite ergodic theory. Probab. Theory Related Fields 135 (2006) 15-52. | MR | Zbl

[21] M. J. Wichura. Functional laws of the iterated logarithm for the partial sums of iid random variables in the domain of attraction of a completely asymmetric stable law. Ann. Probab. 2 (1974) 1108-1138. | MR | Zbl

[22] R. Zweimüller. Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points. Nonlinearity 11 (1998) 1263-1276. | MR | Zbl

[23] R. Zweimüller. Ergodic properties of infinite measure preserving interval maps with indifferent fixed points. Ergodic Theory Dynam. Sys. 20 (2000) 1519-1549. | MR | Zbl

[24] R. Zweimüller. Infinite measure preserving transformations with compact first regeneration. J. Anal. Math. 103 (2007) 93-131. | MR | Zbl

[25] R. Zweimüller. Measure preserving extensions and minimal wandering rates. Israel J. Math. 181 (2011) 295-303. | MR | Zbl

Cited by Sources: