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Laws of large numbers for supercritical branching Gaussian processes

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  • Kouritzin, Michael A.
  • Lê, Khoa
  • Sezer, Deniz

Abstract

A general class of non-Markov, supercritical Gaussian branching particle systems is introduced and its long-time asymptotics is studied. Both weak and strong laws of large numbers are developed with the limit object being characterized in terms of particle motion/mutation. Long memory processes, like branching fractional Brownian motion and fractional Ornstein–Uhlenbeck processes with large Hurst parameters, as well as rough processes, like fractional processes with smaller Hurst parameter, are included as important examples. General branching with second moments is allowed and moment measure techniques are utilized.

Suggested Citation

  • Kouritzin, Michael A. & Lê, Khoa & Sezer, Deniz, 2019. "Laws of large numbers for supercritical branching Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 129(9), pages 3463-3498.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:9:p:3463-3498
    DOI: 10.1016/j.spa.2018.09.011
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    References listed on IDEAS

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    1. Wang, Jiang-Feng & Liang, Han-Ying, 2008. "A note on the almost sure central limit theorem for negatively associated fields," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1964-1970, September.
    2. Chen, Shouquan & Lin, Zhengyan, 2008. "Almost sure functional central limit theorems for weakly dependent sequences," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1683-1693, September.
    3. Blount, Douglas & Kouritzin, Michael A., 2010. "On convergence determining and separating classes of functions," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1898-1907, September.
    4. Li, Yun-Xia & Wang, Jian-Feng, 2008. "An almost sure central limit theorem for products of sums under association," Statistics & Probability Letters, Elsevier, vol. 78(4), pages 367-375, March.
    5. Kouritzin, Michael A. & Ren, Yan-Xia, 2014. "A strong law of large numbers for super-stable processes," Stochastic Processes and their Applications, Elsevier, vol. 124(1), pages 505-521.
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