On the functional central limit theorem and the law of the iterated logarithm for Markov processes

  • R. N. Bhattacharya


Let Xt∶t≧0 be an ergodic stationary Markov process on a state space S. If  is its infinitesimal generator on L2(S, dm), where m is the invariant probability measure, then it is shown that for all f in the range of \(\hat A,n^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \int\limits_0^{nt} {f(X_s ){\text{ }}ds{\text{ }}(t\underline{\underline > } } 0)\) converges in distribution to the Wiener measure with zero drift and variance parameter σ2 =−2〈f, g〉=−2〈Âg, g〉 where g is some element in the domain of  such that Âg=f (Theorem 2.1). Positivity of σ2 is proved for nonconstant f under fairly general conditions, and the range of  is shown to be dense in 1. A functional law of the iterated logarithm is proved when the (2+δ)th moment of f in the range of  is finite for some δ>0 (Theorem 2.7(a)). Under the additional condition of convergence in norm of the transition probability p(t, x, d y) to m(dy) as t → ∞, for each x, the above results hold when the process starts away from equilibrium (Theorems 2.6, 2.7 (b)). Applications to diffusions are discussed in some detail.


State Space Probability Measure Limit Theorem Variance Parameter Markov Process 
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  1. 1.
    Baxter, J.r., Brosamler, G.A.: Energy and the law of the iterated logarithm. Math. Scand. 38, 115–136 (1976)Google Scholar
  2. 2.
    Bensoussan, A., Lions, J.L., Papanicolau: Asymptotic Analysis for Periodic Structures. Amsterdam: North Holland 1978Google Scholar
  3. 3.
    Bhattacharya, R.N.: Criteria for recurrence and existence of invariant probability measures for multidimensional diffusions. Ann. Probab. 6, 541–553 (1978).Google Scholar
  4. 3a.
    Correction Note, Ibid, 8, 1194–95 (1980)Google Scholar
  5. 4.
    Bhattacharaya, R.N., Ramasubramanian, S.: Recurrence and ergodicity of diffusions. J. Multivariate Analysis. (To appear: 1982)Google Scholar
  6. 5.
    Billingsley, P.: The Lindeberg-Lévy theorem for martingales. Proc. Amer. Math. Soc. 12, 788–792 (1961)Google Scholar
  7. 6.
    Billingsley, P.: Convergence of Probability Measures. New York: Wiley 1968Google Scholar
  8. 7.
    Blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. New York: Acad. Press 1968Google Scholar
  9. 8.
    Doob, J.L.: Stochastic Processes. New York: Wiley 1953Google Scholar
  10. 9.
    Dym, H., McKean, H.P., Jr.: Gaussian Processes, Function Theory and the Inverse Spectral Problem. New York: Academic Press 1976Google Scholar
  11. 10.
    Dynkin, E.B.: Markov Processes. Vol. I. Berlin-Heidelberg-New York: Springer 1965Google Scholar
  12. 11.
    Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Berlin-Heidelberg-New York: Springer 1972Google Scholar
  13. 12.
    Gordin, M.I., Lifšic, B.A.: The central limit theorem for stationary Markov processes. Dokl. Akad. Nauk SSSR, 19, 392–394 (1978)Google Scholar
  14. 13.
    Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. New York: Academic Press 1980Google Scholar
  15. 14.
    Ibragimov, I.A.: A central limit theorem for a class of dependent random variables. Theory Probab. Appl. 8, 83–89 (1963)Google Scholar
  16. 15.
    Itô, K., McKean, H.P., Jr.: Diffusion Processes and Their Sample Paths. Berlin-Heidelberg-New York: Springer 1965Google Scholar
  17. 16.
    Khas'minskii, R.A.: Ergodic properties of recurrent diffusion processes and stabilization of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5, 179–196 (1960)Google Scholar
  18. 17.
    Mandl, P.: Analytical Treatment of One Dimensional Markov Processes. Berlin-Heidelberg-New York: Springer 1968Google Scholar
  19. 18.
    McKean, H.P., Jr.: Stochastic Integrals. New York: Academic Press 1969Google Scholar
  20. 19.
    Orey, S.: Limit Theorems for Markov Chain Transition Probabilities. New York: Van Nostrand 1971Google Scholar
  21. 20.
    Rosenblatt, M.: Markov Processes: Structure and Asymptotic Behavior. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  22. 21.
    Yosida, K.: Functional Analysis. Berlin-Heidelberg-New York: Springer 1965Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • R. N. Bhattacharya
    • 1
  1. 1.Dept. of MathematicsIndiana UniversityBloomington

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