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Teor. Veroyatnost. i Primenen., 2002, Volume 47, Issue 4, Pages 727–746 (Mi tvp3777)  

This article is cited in 14 scientific papers (total in 14 papers)

The large deviation principle for stochastic processes. I

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/tvp3777

Full text: PDF file (1780 kB)

English version:
Theory of Probability and its Applications, 2003, 47:4, 567–583

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Received: 05.04.2001
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Citation: M. A. Arcones, “The large deviation principle for stochastic processes. I”, Teor. Veroyatnost. i Primenen., 47:4 (2002), 727–746; Theory Probab. Appl., 47:4 (2003), 567–583

Citation in format AMSBIB
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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  crossref  mathscinet  zmath  isi  scopus
    2. Varron D., “Uniformité en $h$ dans la loi fonctionnelle limite uniforme pour les accroissements du processus empirique indéxé par des fonctions [Uniformity in $h$ in the functional limit law for the increments of the empirical process indexed by functions]”, C. R. Math. Acad. Sci. Paris, 340:6 (2005), 453–456  crossref  mathscinet  zmath  isi  scopus
    3. Varron D., “A nonstandard uniform functional law of the logarithm for the increments of the multivariate empirical process”, C. R. Math. Acad. Sci. Paris, 343:6 (2006), 427–430  crossref  mathscinet  zmath  isi  scopus
    4. Varron D., “Une remarque concernant les principes de grandes déviations dans les espaces Schauder décomposables [A note on large deviation principles in Schauder decomposable spaces]”, C. R. Math. Acad. Sci. Paris, 343:5 (2006), 345–348  crossref  mathscinet  zmath  isi  scopus
    5. Arcones M.A., “Large deviations for M-estimators”, Ann. Inst. Statist. Math., 58:1 (2006), 21–52  crossref  mathscinet  zmath  isi  scopus
    6. 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  crossref  mathscinet  zmath  isi  scopus
    7. 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  mathscinet  zmath  isi
    8. Merlevède F., Peligrad M., “Moderate deviations for linear processes generated by martingale-like random variables”, J. Theor. Probab., 23:1 (2010), 277–300  crossref  mathscinet  zmath  isi  scopus
    9. Maumy M., Varron D., “Non standard functional limit laws for the increments of the compound empirical distribution function”, Electronic Journal of Statistics, 4 (2010), 1324–1344  crossref  mathscinet  zmath  isi  scopus
    10. Nane E., “Stochastic Solutions of a Class of Higher Order Cauchy Problems in $R^d$”, Stoch Dyn, 10:3 (2010), 341–366  crossref  mathscinet  zmath  isi  elib  scopus
    11. Gao F., Zhao X., “Delta Method in Large Deviations and Moderate Deviations for Estimators”, Ann Statist, 39:2 (2011), 1211–1240  crossref  mathscinet  zmath  isi  scopus
    12. Theory Probab. Appl., 59:1 (2015), 70–86  mathnet  crossref  crossref  mathscinet  isi  elib
    13. Cuny Ch., Dedecker J., Merlevede F., “Large and Moderate Deviations For the Left Random Walk on Gl(D)(R)”, ALEA-Latin Am. J. Probab. Math. Stat., 14:1 (2017), 503–527  mathscinet  zmath  isi
    14. Miao Yu., Zhu J., Mu J., “Moderate Deviation Principle For M-Dependent Random Variables()”, Lith. Math. J., 58:1 (2018), 54–68  crossref  mathscinet  zmath  isi  scopus
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