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A central limit theorem for functions of a Markov chain with applications to shifts

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  • Woodroofe, Michael

Abstract

A sufficient condition is developed for partial sums of a function of a stationary, ergodic Markov chain to be asymptotically normal. For Bernoulli and Lebesgue shifts, the condition may be related to the Fourier coefficients of the given function; and the latter condition is shown to be satisfied by most square integrable functions in the case of Bernoulli shifts.

Suggested Citation

  • Woodroofe, Michael, 1992. "A central limit theorem for functions of a Markov chain with applications to shifts," Stochastic Processes and their Applications, Elsevier, vol. 41(1), pages 33-44, May.
    • Handle: RePEc:eee:spapps:v:41:y:1992:i:1:p:33-44
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    Cited by:

    1. Biao Wu, Wei & Min, Wanli, 2005. "On linear processes with dependent innovations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 939-958, June.
    2. L. Ouchti & D. Volný, 2008. "A Conditional CLT which Fails for Ergodic Components," Journal of Theoretical Probability, Springer, vol. 21(3), pages 687-703, September.
    3. Alsmeyer, Gerold & Buckmann, Fabian, 2019. "An arcsine law for Markov random walks," Stochastic Processes and their Applications, Elsevier, vol. 129(1), pages 223-239.
    4. Holzmann, Hajo, 2005. "Martingale approximations for continuous-time and discrete-time stationary Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 115(9), pages 1518-1529, September.
    5. Klicnarová, Jana & Volný, Dalibor, 2009. "On the exactness of the Wu-Woodroofe approximation," Stochastic Processes and their Applications, Elsevier, vol. 119(7), pages 2158-2165, July.
    6. Volný, Dalibor & Woodroofe, Michael, 2014. "Quenched central limit theorems for sums of stationary processes," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 161-167.
    7. Gerold Alsmeyer & Chiranjib Mukherjee, 2023. "On Null-Homology and Stationary Sequences," Journal of Theoretical Probability, Springer, vol. 36(1), pages 1-25, March.
    8. Jérôme Dedecker & Florence Merlevède & Dalibor Volný, 2007. "On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 20(4), pages 971-1004, December.

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