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On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields

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  • Lin, Han-Mai
  • Merlevède, Florence

Abstract

In this paper, we study the weak invariance principle for stationary ortho-martingales with values in 2-smooth or cotype 2 Banach spaces. Then, with the help of a suitable maximal ortho-martingale approximation, we derive the weak invariance principle for stationary random fields in Lp, 1≤p≤2, under a condition in the spirit of Hannan. As an application, we get an asymptotic result for the Lp-distances (1≤p≤2) between the common distribution function and the corresponding empirical distribution function of stationary random fields.

Suggested Citation

  • Lin, Han-Mai & Merlevède, Florence, 2022. "On the weak invariance principle for ortho-martingale in Banach spaces. Application to stationary random fields," Stochastic Processes and their Applications, Elsevier, vol. 153(C), pages 198-220.
    • Handle: RePEc:eee:spapps:v:153:y:2022:i:c:p:198-220
    DOI: 10.1016/j.spa.2022.08.003
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    References listed on IDEAS

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    1. Dedecker, J. & Merlevède, F., 2015. "Moment bounds for dependent sequences in smooth Banach spaces," Stochastic Processes and their Applications, Elsevier, vol. 125(9), pages 3401-3429.
    2. 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.
    3. 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.
    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.
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