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A local limit theorem for linear random fields

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  • Timothy Fortune
  • Magda Peligrad
  • Hailin Sang

Abstract

In this article, we establish a local limit theorem for linear fields of random variables constructed from i.i.d. innovations each with finite second moment. When the coefficients are absolutely summable we do not restrict the region of summation. However, when the coefficients are only square‐summable we add the variables on unions of rectangle and we impose regularity conditions on the coefficients depending on the number of rectangles considered. Our results are new also for the dimension 1, that is, for linear sequences of random variables. The examples include the fractionally integrated processes for which the results of a simulation study is also included.

Suggested Citation

  • Timothy Fortune & Magda Peligrad & Hailin Sang, 2021. "A local limit theorem for linear random fields," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 696-710, September.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:696-710
    DOI: 10.1111/jtsa.12556
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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. Hira Koul & Nao Mimoto & Donatas Surgailis, 2016. "A goodness-of-fit test for marginal distribution of linear random fields with long memory," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 79(2), pages 165-193, February.
    3. Mallik, Atul & Woodroofe, Michael, 2011. "A Central Limit Theorem for linear random fields," Statistics & Probability Letters, Elsevier, vol. 81(11), pages 1623-1626, November.
    4. Peligrad, Magda & Sang, Hailin, 2012. "Asymptotic Properties Of Self-Normalized Linear Processes With Long Memory," Econometric Theory, Cambridge University Press, vol. 28(3), pages 548-569, June.
    5. Wang, Qiying & Lin, Yan-Xia & Gulati, Chandra M., 2001. "Asymptotics for moving average processes with dependent innovations," Statistics & Probability Letters, Elsevier, vol. 54(4), pages 347-356, October.
    6. Maller, R. A., 1978. "A local limit theorem for independent random variables," Stochastic Processes and their Applications, Elsevier, vol. 7(1), pages 101-111, March.
    7. Lahiri, S.N. & Robinson, Peter M., 2016. "Central limit theorems for long range dependent spatial linear processes," LSE Research Online Documents on Economics 65331, London School of Economics and Political Science, LSE Library.
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    Cited by:

    1. Peligrad, Magda & Sang, Hailin & Zhang, Na, 2024. "On the local limit theorems for linear sequences of lower psi-mixing Markov chains," Statistics & Probability Letters, Elsevier, vol. 210(C).
    2. Peligrad, Magda & Sang, Hailin & Xiao, Yimin & Yang, Guangyu, 2022. "Limit theorems for linear random fields with innovations in the domain of attraction of a stable law," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 596-621.

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