Central limit theorems for non-invertible measure preserving maps

Colloquium Mathematicum

Volumn 110, Issue 1, 2008, Pages 167-191

  Mackey, Michael C ^a   Tyran Kaminńska, Marta ^b  

^a MCGILL UNIVERSITY   (Canada)

^b UNIVERSITY OF SILESIA   (Poland)

Author keywords

Asymptotic periodicity; Functional central limit theorem; Maximal inequality; Measure preserving transformation; Perron frobenius operator; Tent map

Indexed keywords

EID: 77953147662     PISSN: 00101354     EISSN: 17306302     Source Type: Journal    
DOI: 10.4064/cm110-1-6     Document Type: Article

References (35)

1. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
2. P. Billingsley, Probability and Measure, Wiley Ser. Probab. Math. Statist., Wiley, New York, 1995.
3. A. Boyarsky and M. Scarowsky, On a class of transformations which have unique absolutely continuous invariant measures, Trans. Amer. Math. Soc. 255 (1979), 243-262.
4. J.-P. Conze and S. Le Borgne, Methode de martingales et flot geodesique sur une surface de courbure constante negative, Ergodic Theory Dynarn. Systems 21 (2001), 421-441.
5. M. Denker, The central limit theorem for dynamical systems, in: Dynamical Systems and Ergodic Theory (Warszawa, 1986), Banach Center Publ. 23, PWN, Warszawa, 1989, 33-62.
6. G. K. Eagleson, Martingale convergence to mixtures of infinitely divisible laws, Ann. Probab. 3 (1975), 557-562.
7. M. I. Gordin, The central limit theorem for stationary processes, Dokl. Akad. Nauk SSSR 188 (1969), 739-741 (in Russian); English transl.: Soviet Math. Dokl. 10 (1969), 1174-1176.
8. S. Gouëzel, Central limit theorem and stable laws for intermittent maps, Probab. Theory Related Fields 128 (2004), 82-122.
9. T. Inoue and H. Ishitani, Asymptotic periodicity of densities and ergodic properties for nonsingular systems, Hiroshima Math. J. 21 (1991), 597-620.
10. C. T. Ionescu Tulcea and G. Marinescu, Théorie ergodique pour des classes d’opérations non complètement continues, Ann. of Math. (2) 52 (1950), 140-147.
11. S. Ito, S. Tanaka and H. Nakada, On unimodal linear transformations and chaos, I, Tokyo J. Math. 2 (1979), 221-239.
12. M. Jabłorński and J. Malczak, A central limit theorem for piecewise convex mappings of the unit interval, Tôhoku Math. J. (2) 35 (1983), 173-180.
13. G. Keller, Un théorème de la limite centrale pour une classe de transformations monotones par morceaux, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), A155-A158.
14. J. Komorník and A. Lasota, Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35 (1987), 321-327.
15. A. Lasota and M. C. Mackey, Chaos, Fractals, and Noise, Appl. Math. Sci. 97, Springer, New York, 1994.
16. C. Liverani, Central limit theorem for deterministic systems, in: Internat. Conf. on Dynamical Systems (Montevideo, 1995), Pitman Res. Notes Math. Ser. 362, Longman, Harlow, 1996, 56-75.
17. M. C. Mackey and M. Tyran-Kamińska, Deterministic Brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system, Phys. Rep. 422 (2006), 167-222.
18. M. Maxwell and M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000), 713-724.
19. I. Melbourne and M. Nicol, Statistical properties of endomorphisms and compact group extensions, J. London Math. Soc. (2) 70 (2004), 427-446.
20. I. Melbourne and A. Török, Central limit theorems and invariance principles for time-one maps of hyperbolic flows, Comm. Math. Phys. 229 (2002), 57-71.
21. F. Merlevède, M. Peligrad and S. Utev, Recent advances in invariance principles for stationary sequences, Probab. Surv. 3 (2006), 1-36 (electronic).
22. M. Peligrad and S. Utev, A new maximal inequality and invariance principle for stationary sequences, Ann. Probab. 33 (2005), 798-815.
23. M. Peligrad, S. Utev and W. B. Wu, A maximal Lp-inequality for stationary sequences and its applications, Proc. Amer. Math. Soc. 135 (2007), 541-550.
24. N. Provatas and M. C. Mackey, Asymptotic periodicity and banded chaos, Phys. D 53 (1991), 295-318.
25. J. Rousseau-Egele, Un théorème de la limite locale pour une classe de transformations dilatantes et monotones par morceaux, Ann. Probab. 11 (1983), 772-788.
26. M. Rychlik, Bounded variation and invariant measures, Studia Math. 76 (1983), 69-80.
27. M. Tyran-Kamińska, An invariance principle for maps with polynomial decay of correlations, Comm. Math. Phys. 260 (2005), 1-15.
28. D. Volný, A nonergodic version of Gordin’s CLT for integrable stationary processes, Comment. Math. Univ. Carolin. 28 (1987), 413-419.
29. D. Volný, On the invariance principle and functional law of iterated logarithm for nonergodic processes, Yokohama Math. J. 35 (1987), 137-141.
30. D. Volný, On nonergodic versions of limit theorems, Appl. Mat. 34 (1989), 351-363.
31. D. Volný, Approximating martingales and the central limit theorem for strictly stationary processes, Stochastic Process. Appl. 44 (1993), 41-74.
32. S. Wong, A central limit theorem for piecewise monotonic mappings of the unit interval, Ann. Probab. 7 (1979), 500-514.
33. T. Yoshida, H. Mori and H. Shigematsu, Analytic study of chaos of the tent map: band structures, power spectra, and critical behaviors, J. Statist. Phys. 31 (1983), 279-308.
34. L.-S. Young, Recurrence times and rates of mixing, Israel J. Math. 110 (1999), 153-188.
35. R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity 11 (1998), 1263-1276.