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Approximating martingales and the central limit theorem for strictly stationary processes

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  • Volný, Dalibor

Abstract

The proofs of various central limit theorems for strictly stationary sequences of random variables are based on approximating the partial sums of the process by martingales (cf., e.g., Gordin, 1969; Dürr and Goldstein, 1984; or Hall and Heyde, 1980, Chapter 5). Here we shall give a study on the assumptions of such theorems and introduce new ones. Then we shall discuss conditions under which the results take place in almost all ergodic components simultaneously and present an application to the limit theory of stationary linear proceses with random coefficients.

Suggested Citation

  • Volný, Dalibor, 1993. "Approximating martingales and the central limit theorem for strictly stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 44(1), pages 41-74, January.
    • Handle: RePEc:eee:spapps:v:44:y:1993:i:1:p:41-74
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    Citations

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    Cited by:

    1. Yokoyama, Ryozo, 1995. "On the central limit theorem and law of the iterated logarithm for stationary processes with applications to linear processes," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 343-351, October.
    2. Cuny, Christophe & Fan, Ai Hua, 2017. "Study of almost everywhere convergence of series by mean of martingale methods," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2725-2750.
    3. Lesigne, Emmanuel & Volný, Dalibor, 2001. "Large deviations for martingales," Stochastic Processes and their Applications, Elsevier, vol. 96(1), pages 143-159, November.
    4. Dede, Sophie, 2009. "An empirical Central Limit Theorem in for stationary sequences," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3494-3515, October.
    5. Magda Peligrad & Hailin Sang, 2013. "Central Limit Theorem for Linear Processes with Infinite Variance," Journal of Theoretical Probability, Springer, vol. 26(1), pages 222-239, March.
    6. Peligrad, Magda, 2020. "A new CLT for additive functionals of Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 130(9), pages 5695-5708.
    7. Dedecker, Jérôme & Merlevède, Florence, 2011. "Rates of convergence in the central limit theorem for linear statistics of martingale differences," Stochastic Processes and their Applications, Elsevier, vol. 121(5), pages 1013-1043, May.
    8. Wu, Wei Biao, 2009. "An asymptotic theory for sample covariances of Bernoulli shifts," Stochastic Processes and their Applications, Elsevier, vol. 119(2), pages 453-467, February.
    9. 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.
    10. Volný, Dalibor & Wang, Yizao, 2014. "An invariance principle for stationary random fields under Hannan’s condition," Stochastic Processes and their Applications, Elsevier, vol. 124(12), pages 4012-4029.
    11. Dalibor Volný, 2010. "Martingale Approximation and Optimality of Some Conditions for the Central Limit Theorem," Journal of Theoretical Probability, Springer, vol. 23(3), pages 888-903, September.

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