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Hitting and returning to rare events for all alpha-mixing processes

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  • Abadi, Miguel
  • Saussol, Benoit

Abstract

We prove that for any [alpha]-mixing stationary process the hitting time of any n-string An converges, when suitably normalized, to an exponential law. We identify the normalization constant [lambda](An). A similar statement holds also for the return time. To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem of Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any n-string in n consecutive observations goes to zero as n goes to infinity.

Suggested Citation

  • Abadi, Miguel & Saussol, Benoit, 2011. "Hitting and returning to rare events for all alpha-mixing processes," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 314-323, February.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:2:p:314-323
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    Cited by:

    1. Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes & Todd, Mike, 2015. "Speed of convergence for laws of rare events and escape rates," Stochastic Processes and their Applications, Elsevier, vol. 125(4), pages 1653-1687.
    2. Freitas, Ana Cristina Moreira & Freitas, Jorge Milhazes & Todd, Mike & Vaienti, Sandro, 2016. "Rare events for the Manneville–Pomeau map," Stochastic Processes and their Applications, Elsevier, vol. 126(11), pages 3463-3479.

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