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Quenched Invariance Principles for Orthomartingale-Like Sequences

Author

Listed:
  • Magda Peligrad

    (University of Cincinnati)

  • Dalibor Volný

    (LMRS, CNRS and Université de Rouen Normandie)

Abstract

In this paper, we study the central limit theorem and its functional form for random fields which are started not from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so-called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.

Suggested Citation

  • Magda Peligrad & Dalibor Volný, 2020. "Quenched Invariance Principles for Orthomartingale-Like Sequences," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1238-1265, September.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:3:d:10.1007_s10959-019-00914-z
    DOI: 10.1007/s10959-019-00914-z
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    References listed on IDEAS

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    1. 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.
    2. Volný, Dalibor & Woodroofe, Michael, 2014. "Quenched central limit theorems for sums of stationary processes," Statistics & Probability Letters, Elsevier, vol. 85(C), pages 161-167.
    3. 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.
    4. 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.
    5. 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.
    6. Volný, Dalibor, 2019. "On limit theorems for fields of martingale differences," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 841-859.
    7. 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.
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    Cited by:

    1. Giraudo, Davide, 2024. "Deviation inequality for Banach-valued orthomartingales," Stochastic Processes and their Applications, Elsevier, vol. 175(C).

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