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An almost sure limit theorem for the maxima of smooth stationary Gaussian processes

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  • Tan, Zhongquan

Abstract

Let {X(t),t≥0} be a continuous mean square differentiable stationary Gaussian process. Under some mild restrictions on its correlation function r(⋅), we prove an almost sure limit theorem for the maximum of the Gaussian process {X(t),t≥0}.

Suggested Citation

  • Tan, Zhongquan, 2013. "An almost sure limit theorem for the maxima of smooth stationary Gaussian processes," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 2135-2141.
    • Handle: RePEc:eee:stapro:v:83:y:2013:i:9:p:2135-2141
    DOI: 10.1016/j.spl.2013.05.034
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    References listed on IDEAS

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    1. Arendarczyk, Marek & Dȩbicki, Krzysztof, 2012. "Exact asymptotics of supremum of a stationary Gaussian process over a random interval," Statistics & Probability Letters, Elsevier, vol. 82(3), pages 645-652.
    2. Fahrner, I. & Stadtmüller, U., 1998. "On almost sure max-limit theorems," Statistics & Probability Letters, Elsevier, vol. 37(3), pages 229-236, March.
    3. Csáki, Endre & Gonchigdanzan, Khurelbaatar, 2002. "Almost sure limit theorems for the maximum of stationary Gaussian sequences," Statistics & Probability Letters, Elsevier, vol. 58(2), pages 195-203, June.
    4. De[combining cedilla]bicki, Krzysztof & Tabis, Kamil, 2011. "Extremes of the time-average of stationary Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2049-2063, September.
    5. Hashorva, Enkelejd & Weng, Zhichao, 2013. "Limit laws for extremes of dependent stationary Gaussian arrays," Statistics & Probability Letters, Elsevier, vol. 83(1), pages 320-330.
    6. Tan, Zhongquan & Peng, Zuoxiang, 2009. "Almost sure convergence for non-stationary random sequences," Statistics & Probability Letters, Elsevier, vol. 79(7), pages 857-863, April.
    7. Bercu, B., 2004. "On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications," Stochastic Processes and their Applications, Elsevier, vol. 111(1), pages 157-173, May.
    8. Hösler, Jörg & Piterbarg, Vladimir & Rumyantseva, Ekaterina, 2011. "Extremes of Gaussian processes with a smooth random variance," Stochastic Processes and their Applications, Elsevier, vol. 121(11), pages 2592-2605, November.
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