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Large deviations for the local fluctuations of random walks


  • Barral, Julien
  • Loiseau, Patrick


We establish large deviation properties valid for almost every sample path of a class of stationary mixing processes (X1,...,Xn,...). These properties are inherited from those of and describe how the local fluctuations of almost every realization of Sn deviate from the almost sure behavior. These results apply to the fluctuations of Brownian motion, Birkhoff averages on hyperbolic dynamics, as well as branching random walks. Also, they lead to new insights into the "randomness" of the digits of expansions in integer bases of Pi. We formulate a new conjecture, supported by numerical experiments, implying the normality of Pi.

Suggested Citation

  • Barral, Julien & Loiseau, Patrick, 2011. "Large deviations for the local fluctuations of random walks," Stochastic Processes and their Applications, Elsevier, vol. 121(10), pages 2272-2302, October.
  • Handle: RePEc:eee:spapps:v:121:y:2011:i:10:p:2272-2302

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    References listed on IDEAS

    1. Devroye, Luc & Fawzi, Omar, 2010. "Simulating the Dickman distribution," Statistics & Probability Letters, Elsevier, vol. 80(3-4), pages 242-247, February.
    2. de Haan, Laurens & Resnick, Sidney I. & Rootzén, Holger & de Vries, Casper G., 1989. "Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes," Stochastic Processes and their Applications, Elsevier, vol. 32(2), pages 213-224, August.
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