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On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria

Author

Listed:
  • Na Zhang

    (Towson University)

  • Lucas Reding

    (Université de Rouen Normandie)

  • Magda Peligrad

    (University of Cincinnati)

Abstract

Motivated by random evolutions which do not start from equilibrium, in a recent work, Peligrad and Volný (J Theor Probab, 2018. arXiv:1802.09106 ) showed that the central limit theorem (CLT) holds for stationary ortho-martingale random fields when they are started from a fixed past trajectory. In this paper, we study this type of behavior, also known under the name of quenched CLT, for a class of random fields larger than the ortho-martingales. We impose sufficient conditions in terms of projective criteria under which the partial sums of a stationary random field admit an ortho-martingale approximation. More precisely, the sufficient conditions are of the Hannan’s projective type. We also discuss some aspects of the functional form of the quenched CLT. As applications, we establish new quenched CLTs and their functional form for linear and nonlinear random fields with independent innovations.

Suggested Citation

  • Na Zhang & Lucas Reding & Magda Peligrad, 2020. "On the Quenched Central Limit Theorem for Stationary Random Fields Under Projective Criteria," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2351-2379, December.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:4:d:10.1007_s10959-019-00943-8
    DOI: 10.1007/s10959-019-00943-8
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    References listed on IDEAS

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    1. El Machkouri, Mohamed & Volný, Dalibor & Wu, Wei Biao, 2013. "A central limit theorem for stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 123(1), pages 1-14.
    2. Christophe Cuny & Magda Peligrad, 2012. "Central Limit Theorem Started at a Point for Stationary Processes and Additive Functionals of Reversible Markov Chains," Journal of Theoretical Probability, Springer, vol. 25(1), pages 171-188, March.
    3. Volný, Dalibor & Woodroofe, Michael, 2014. "Quenched central limit theorems for sums of stationary processes," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 161-167.
    4. Barrera, David & Peligrad, Costel & Peligrad, Magda, 2016. "On the functional CLT for stationary Markov chains started at a point," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1885-1900.
    5. Peligrad, Magda & Zhang, Na, 2018. "On the normal approximation for random fields via martingale methods," Stochastic Processes and their Applications, Elsevier, vol. 128(4), pages 1333-1346.
    6. 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.
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    Cited by:

    1. Davydov, Youri & Tempelman, Arkady, 2024. "Randomized limit theorems for stationary ergodic random processes and fields," Stochastic Processes and their Applications, Elsevier, vol. 174(C).

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