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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 1, Pages 122–150 (Mi tvp304)

The large deviation principle for stochastic processes. II

M. A. Arcones

State University of New York, Department of Mathematical Sciences

Abstract: We discuss the large deviation principle of stochastic processes as random elements of $l_{\infty}(T)$. We show that the large deviation principle in $l_{\infty}(T)$ is equivalent to the large deviation principle of the finite dimensional distributions plus an exponential asymptotic equicontinuity condition with respect to a pseudometric which makes $T$ a totally bounded pseudometric space. This result allows us to obtain necessary and sufficient conditions for the large deviation principle of different types of stochastic processes. We discuss the large deviation principle of Gaussian and Poisson processes. As an application, we determine the integrability of the iterated fractional Brownian motion.

Keywords: large deviations, stochastic processes, Gaussian processes, iterated Brownian motion, Poisson process.

DOI: https://doi.org/10.4213/tvp304

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English version:
Theory of Probability and its Applications, 2004, 48:1, 19–44

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Citation: M. A. Arcones, “The large deviation principle for stochastic processes. II”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 122–150; Theory Probab. Appl., 48:1 (2004), 19–44

Citation in format AMSBIB
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• https://doi.org/10.4213/tvp304
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This publication is cited in the following articles:
1. Mason D.M., “A uniform functional law of the logarithm for the local empirical process”, Ann. Probab., 32:2 (2004), 1391–1418
2. Varron D., “A limited in bandwidth uniformity for the functional limit law of the increments of the empirical process”, Electronic Journal of Statistics, 2 (2008), 1043–1064
3. Merlevede F., Peligrad M., “Functional moderate deviations for triangular arrays and applications”, Alea-Latin American Journal of Probability and Mathematical Statistics, 5 (2009), 3–20
4. Merlevède F., Peligrad M., “Moderate deviations for linear processes generated by martingale-like random variables”, J. Theor. Probab., 23:1 (2010), 277–300
5. Coiffard C., “Random Fractals Generated by a Local Gaussian Process Indexed by a Class of Functions”, ESAIM-Probability and Statistics, 15 (2011), 249–269
6. Gao F., Zhao X., “Delta Method in Large Deviations and Moderate Deviations for Estimators”, Ann Statist, 39:2 (2011), 1211–1240
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8. Penda S. Valere Bitseki, Djellout H., Proia F., “Moderate Deviations For the Durbin-Watson Statistic Related To the First-Order Autoregressive Process”, ESAIM-Prob. Stat., 18 (2014), 308–331
9. M. S. Ermakov, “On conistent hypothesis testing”, J. Math. Sci. (N. Y.), 225:5 (2017), 751–769
10. Del Moral P., Hu Sh., Wu L., “Moderate Deviations For Interacting Processes”, Stat. Sin., 25:3 (2015), 921–951
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