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An almost sure central limit theorem for the stochastic heat equation

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  • Li, Jingyu
  • Zhang, Yong

Abstract

Let u(t,x) be the solution to the stochastic heat equation on R+×Rd driven by a Gaussian noise that is white in time and has a spatially covariance that satisfies Dalang’s condition. In this paper, we prove an almost sure central limit theorem for spatial averages of the form ∫[0,N]dg(u(t,x))dx as N→∞ for fixed t>0, where g is a globally Lipschitz function or belongs to a class of locally Lipschitz functions.

Suggested Citation

  • Li, Jingyu & Zhang, Yong, 2021. "An almost sure central limit theorem for the stochastic heat equation," Statistics & Probability Letters, Elsevier, vol. 177(C).
    • Handle: RePEc:eee:stapro:v:177:y:2021:i:c:s0167715221001115
    DOI: 10.1016/j.spl.2021.109149
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    References listed on IDEAS

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    1. Peligrad, Magda & Shao, Qi-Man, 1995. "A note on the almost sure central limit theorem for weakly dependent random variables," Statistics & Probability Letters, Elsevier, vol. 22(2), pages 131-136, February.
    2. Huang, Jingyu & Nualart, David & Viitasaari, Lauri, 2020. "A central limit theorem for the stochastic heat equation," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7170-7184.
    3. Lacey, Michael T. & Philipp, Walter, 1990. "A note on the almost sure central limit theorem," Statistics & Probability Letters, Elsevier, vol. 9(3), pages 201-205, March.
    4. Berkes, István & Csáki, Endre, 2001. "A universal result in almost sure central limit theory," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 105-134, July.
    5. Yong Zhang, 2020. "Further research on limit theorems for self-normalized sums," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 49(2), pages 385-402, January.
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