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Uniform large and moderate deviations for functional empirical processes

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  • Dembo, Amir
  • Zajic, Tim

Abstract

For {Xi}i >= 1 a sequence of i.i.d. random variables taking values in a Polish space [Sigma] with distribution [mu], we obtain large and moderate deviation principles for the processes {n-1 [Sigma][nt]i = 1 [delta]Xi; t >= 0}n >= 1 and {n-1/2 [Sigma][nt]i = 1 ([delta]Xi - [mu]); t >= 0}n >= 1, respectively. Given a class of bounded functions F on [Sigma], we then consider the above processes as taking values in the Banach space of bounded functionals over F and obtain the corresponding (uniform over F), large and moderate deviation principles. Among the corollaries considered are functional laws of the iterated logarithm.

Suggested Citation

  • Dembo, Amir & Zajic, Tim, 1997. "Uniform large and moderate deviations for functional empirical processes," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 195-211, May.
    • Handle: RePEc:eee:spapps:v:67:y:1997:i:2:p:195-211
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    References listed on IDEAS

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    1. Dembo, Amir & Zajic, Tim, 1995. "Large deviations: From empirical mean and measure to partial sums process," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 191-224, June.
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    Cited by:

    1. Kurbanmuradov O. & Sabelfeld K., 2006. "Exponential bounds for the probability deviations of sums of random fields," Monte Carlo Methods and Applications, De Gruyter, vol. 12(3), pages 211-229, October.
    2. Eichelsbacher, Peter & Schmock, Uwe, 1998. "Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem," Stochastic Processes and their Applications, Elsevier, vol. 77(2), pages 233-251, September.
    3. Klebaner, F. C. & Liptser, R., 1999. "Moderate deviations for randomly perturbed dynamical systems," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 157-176, April.
    4. Djellout, Hacène & Guillin, Arnaud & Samoura, Yacouba, 2017. "Estimation of the realized (co-)volatility vector: Large deviations approach," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2926-2960.

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    60F10 60B12 60G50;

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