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Laws of the iterated logarithm for a class of iterated processes

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  • Nane, Erkan

Abstract

Let X={X(t),t>=0} be a Brownian motion or a spectrally negative stable process of index 1 =0} be the hitting time of a stable subordinator of index 0 0. This establishes the lower bound in the law of the iterated logarithm which we could not prove with the techniques of our paper [Meerschaert, M.M., Nane, E., Xiao, Y.. 2008. Large deviations for local time fractional Brownian motion and applications. J. Math. Anal. Appl. 346, 432-445]. We also obtain exact small ball probability for X(E(t)) using ideas from Aurzada and Lifshits [Aurzada, F., Lifshits, M., On the small deviation problem for some iterated processes. preprint: arXiv:0806.2559].

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  • Nane, Erkan, 2009. "Laws of the iterated logarithm for a class of iterated processes," Statistics & Probability Letters, Elsevier, vol. 79(16), pages 1744-1751, August.
    • Handle: RePEc:eee:stapro:v:79:y:2009:i:16:p:1744-1751
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2009. "Correlated continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 79(9), pages 1194-1202, May.
    2. Csáki, Endre & Csörgo, Miklós & Földes, Antónia & Révész, Pál, 1995. "Global Strassen-type theorems for iterated Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 59(2), pages 321-341, October.
    3. Nane, Erkan, 2008. "Isoperimetric-type inequalities for iterated Brownian motion in," Statistics & Probability Letters, Elsevier, vol. 78(1), pages 90-95, January.
    4. Bertoin, Jean, 1996. "Iterated Brownian motion and stable() subordinator," Statistics & Probability Letters, Elsevier, vol. 27(2), pages 111-114, April.
    5. Nane, Erkan, 2006. "Iterated Brownian motion in bounded domains in," Stochastic Processes and their Applications, Elsevier, vol. 116(6), pages 905-916, June.
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    Cited by:

    1. Iksanov, Alexander & Kabluchko, Zakhar & Marynych, Alexander & Shevchenko, Georgiy, 2017. "Fractionally integrated inverse stable subordinators," Stochastic Processes and their Applications, Elsevier, vol. 127(1), pages 80-106.
    2. Endre Csáki & Antónia Földes, 2020. "Random Walks on Comb-Type Subsets of $$\mathbb {Z}^2$$ Z 2," Journal of Theoretical Probability, Springer, vol. 33(4), pages 2233-2257, December.
    3. Gajda, Janusz & Magdziarz, Marcin, 2014. "Large deviations for subordinated Brownian motion and applications," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 149-156.
    4. Kobayashi, Kei, 2016. "Small ball probabilities for a class of time-changed self-similar processes," Statistics & Probability Letters, Elsevier, vol. 110(C), pages 155-161.

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