Abstract
We show that distributional and weak functional limit theorems for ergodic processes often hold for arbitrary absolutely continuous initial distributions. This principle is illustrated in the setup of ergodic sums, renewal-theoretic variables, and hitting times for ergodic measure preserving transformations.
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References
Aaronson, J.: The asymptotic distributional behaviour of transformations preserving infinite measures. J. Anal. Math. 39, 203–234 (1981)
Aaronson, J.: Random f-expansions. Ann. Probab. 14, 1037–1057 (1986)
Aaronson, J.: An Introduction to Infinite Ergodic Theory. Am. Math. Soc., Providence (1997)
Abadi, M., Galves, A.: Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov. Process. Relat. Fields 7, 97–112 (2001)
Aldous, D.J., Eagleson, G.K.: On mixing and stability of limit theorems. Ann. Probab. 6, 325–331 (1978)
Billingsley, P.: Convergence of Probability Measures, 2nd edn. Wiley, New York (1999)
Bressaud, X., Zweimüller, R.: Non exponential law of entrance times in asymptotically rare events for intermittent maps with infinite invariant measure. Ann. H. Poincaré 2, 501–512 (2001)
Chazottes, J.-R., Gouëzel, S.: On almost-sure versions of classical limit theorems for dynamical systems. Probab. Theory Relat. Fields 138, 195–234 (2007)
Collet, P., Galves, A.: Statistics of close visits to the indifferent fixed point of an interval map. J. Stat. Phys. 72, 459–478 (1993)
Darling, D.A.: The influence of the maximal term in the addition of independent random variables. Trans. Am. Math. Soc. 73, 95–107 (1952)
Dynkin, E.B.: Some limit theorems for sums of independent random variables with infinite mathematical expectation. Sel. Transl. Math. Stat. Probab. 1, 171–189 (1961)
Eagleson, G.K.: Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21, 653–660 (1976)
Hirata, M.: Poisson law for Axiom A diffeomorphisms. Ergod. Theory Dyn. Syst. 13, 533–556 (1993)
Hirata, M., Saussol, B., Vaienti, S.: Statistics of return times: a general framework and new applications. Commun. Math. Phys. 206, 33–55 (1999)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer, New York (1987)
Karr, A.F.: Point Processes and their Statistical Inference. Dekker, New York (1986)
Krengel, U.: Ergodic Theorems. De Gruyter, Berlin (1985)
Kupsa, M., Lacroix, Y.: Asymptotics for hitting times. Ann. Probab. 33, 610–619 (2005)
Lamperti, J.: Some limit theorems for stochastic processes. J. Math. Mech. 7, 433–448 (1958)
Lamperti, J.: An invariance principle in renewal theory. Ann. Math. Stat. 33, 685–696 (1962)
Lin, M.: Mixing for Markov operators. Z. Wahrsch. verw. Geb. 19, 231–243 (1971)
Melbourne, I., Török, A.: Statistical limit theorems for suspension flows. Israel J. Math. 144, 191–209 (2004)
Pollard, D.: Convergence of Stochastic Processes. Springer, Berlin (1984)
Rényi, A.: On mixing sequences of sets. Acta Math. Acad. Sci. Hung. 9, 3–4 (1958)
Skorohod, A.V.: Limit theorems for random processes. Teor. Verojatnost. i Primenen. 1, 289–319 (1956)
Thaler, M.: The Dynkin-Lamperti arc-sine laws for measure preserving transformations. Trans. Am. Math. Soc. 350, 4593–4607 (1998)
Thaler, M., Zweimüller, R.: Distributional limit theorems in infinite ergodic theory. Probab. Theory Relat. Fields 135, 15–52 (2006)
Wojtaszczyk, P.: Banach Spaces for Analysts. Cambridge Univ. Press (1991)
Yosida, K.: Mean ergodic theorem in Banach spaces. Proc. Imp. Acad. Tokyo 14, 292–294 (1938)
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Zweimüller, R. Mixing Limit Theorems for Ergodic Transformations. J Theor Probab 20, 1059–1071 (2007). https://doi.org/10.1007/s10959-007-0085-y
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DOI: https://doi.org/10.1007/s10959-007-0085-y