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On weak convergence in dynamical systems to self-similar processes with spectral representation

Author: Michael T. Lacey
Journal: Trans. Amer. Math. Soc. 328 (1991), 767-778
MSC: Primary 60F17; Secondary 28D05, 60F05
MathSciNet review: 1066446
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Abstract: Let $ (X,\mu,T)$ be an aperiodic dynamical system. Set $ {S_m}f = f + \cdots + f \circ {T^{m - 1}}$, where $ f$ is a measurable function on $ X$. Let $ Y(t)$ be one of a class of self-similar process with a "nice" spectral representation, for instance, either a fractional Brownian motion, a Hermite process, or a harmonizable fractional stable motion. We show that there is an $ f$ on $ X$, and constants $ {A_m} \to + \infty $ so that

$\displaystyle A_m^{ - 1}{S_{[mt]}}f\mathop \Rightarrow \limits^d Y(t),$

the convergence being understood in the sense of weak convergence of all finite dimensional distributions in $ t$.

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  • [BD] R. Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc. 302 (1987), 715-726. MR 891642 (88i:60039)
  • [CM] S. Cambanis and M. Maejima, Two classes of self-similar stable processes with stationary increments, Stochastic Process. Appl. 32 (1989), 305-329. MR 1014456 (91h:60040)
  • [D] M. Denker, The Central Limit Theorem for dynamical systems, Dynamical Systems and Ergodic Theory, Banach Center Publications, vol. 23, Warsaw, 1989. MR 1102700 (92d:28007)
  • [DK] M. Denker and G. Keller, Rigorous statistical procedures for data from dynamical systems, J. Statist. Phys. 44 (1986), 67-93. MR 854400 (87k:58149)
  • [Dob] R. L. Dobrushin, Gaussian and their subordinated self-similar random generalized fields, Ann. Probab. 7 (1979), 1-28. MR 515810 (80e:60069)
  • [GS] L. Giraitis and D. Surgailis, $ CLT$ and other limit theorems for functionals of Gaussian processes, Z. Wahrsch. Verw. Gebiete 70 (1985), 191-212. MR 799146 (86j:60067)
  • [GL] M. I. Gordin and B. A. Lifsic, The central limit theorem for stationary Markov processes, J. Soviet Math. 19 (1978), 392-394. MR 0501277 (58:18672)
  • [H] C. D. Hardin, On the spectral representation of symmetric stable processes, J. Multivariate Anal. 12 (1982), 385-401. MR 666013 (84g:60121)
  • [LP] M. Lacey and W. Philipp, A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201-205. MR 1045184 (91e:60100)
  • [Lam] J. W. Lamperti, Semistable stochastic processes, Trans. Amer. Math. Soc. 104 (1962), 62-78. MR 0138128 (25:1575)
  • [M] M. Maejima, Self-similar processes and limit theorems, Sugaku Expositions 2 (1989), 103-123. MR 944886 (89j:60070)
  • [Maj] P. Major, Multiple Wiener-Itô Integrals, Lecture Notes in Math., vol. 849, Springer-Verlag, Berlin and New York, 1981. MR 611334 (82i:60099)
  • [MvN] B. B. Mandelbrot and J. W. van Ness, Fractional Brownian motion, fractional noises and applications, SIAM Rev. 10 (1968), 422-437. MR 0242239 (39:3572)
  • [O] H. Oodaira, On Strassen's version of the law of the iterated logarithm for Gaussian processes, Z. Wahrsch. Verw. Gebiete 21 (1972), 289-299. MR 0309181 (46:8291)
  • [Or] J. Ortega, Upper classes for the increments of fractional Wiener processes, Probab. Theory Related Fields 80 (1989), 365-379. MR 976531 (90a:60062)
  • [PS] W. Philipp and W. Stout, Almost sure invariance principles for partial sums of weakly dependent random variables, Mem. Amer. Math. Soc. no. 161, 1975. MR 0433597 (55:6570)
  • [PUZ] F. Przytycki, M. Urbanski, and A. Zdunik, Harmonic, Gibbs and Hausdorff measures on repellers for holomorphic maps, Ann. of Math. (2) 130 (1989), 1-40. MR 1005606 (91i:58115)
  • [R] M. Rosenblatt, Independence and dependence, Proc. 4th Berkeley Sympos. Math. Statist. Probab., Univ. of California Press, Berkeley, Calif., 1961, pp. 411-443. MR 0133863 (24:A3687)
  • [PT] W. Parry and S. Tuncel, Classification problems in ergodic theory, Cambridge Univ. Press, London and New York, 1982. MR 666871 (84g:28024)
  • [S] Ya. G. Sinai, The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. 1 (1960), 983-986. MR 0125607 (23:A2906)
  • [T1] M. S. Taqqu, Random processes with long range dependence and high variability, J. Geophys. Res. 92 (1987), 9683-9686.
  • [T2] -, Self-similar processes and related ultraviolet and infrared catastrophes, Random Fields: Rigorous Results in Statistical Mechanics and Quantum Field Theory, Colloquia Mathematica Societatis Janos Bolyai, vol. 27, Book 2, North-Holland, Amsterdam, 1981, pp. 1057-1096. MR 712727 (84m:82028)
  • [T3] -, Weak convergence to fractional Brownian motion and to the Rosenblatt process, Z. Wahrsch. Verw. Gebiete 31 (1975), 287-302. MR 0400329 (53:4164)
  • [TC] -, A survey of functional laws of the iterated logarithm for self-similar process, Stochastic Models 1 (1985), 77-115. MR 793477 (86m:60091)
  • [Z] A. Zygmund, Trigonometric series, (2nd ed.), Cambridge Univ. Press, London and New York, 1959. MR 0107776 (21:6498)

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Keywords: Aperiodic dynamical system, weak convergence, almost sure invariance principle, spectral representation, self-similar, fractional Brownian motion, Hermite processes, harmonizable fractional stable motion
Article copyright: © Copyright 1991 American Mathematical Society

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