# Large deviations for the local fluctuations of random walks

Abstract : We establish large deviation properties valid for almost every sample path of a class of stationary mixing processes $(X_1,...,X_n,...)$. These properties are inherited from those of $s_n = \sum_{i=1}^n X_i$ 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.
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Article dans une revue
Stochastic Processes and their Applications, Elsevier, 2011, 121 (10), pp.2272-2302. 〈10.1016/j.spa.2011.06.004〉
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https://hal.inria.fr/hal-00844817
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Soumis le : mardi 16 juillet 2013 - 08:17:37
Dernière modification le : jeudi 11 janvier 2018 - 06:12:25
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1004.3713v2.pdf
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Julien Barral, Patrick Loiseau. Large deviations for the local fluctuations of random walks. Stochastic Processes and their Applications, Elsevier, 2011, 121 (10), pp.2272-2302. 〈10.1016/j.spa.2011.06.004〉. 〈hal-00844817〉

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