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Sibirsk. Mat. Zh., 2010, Volume 51, Number 6, Pages 1251–1269 (Mi smj2159)  

This article is cited in 18 scientific papers (total in 18 papers)

On large deviation principles in metric spaces

A. A. Borovkovab, A. A. Mogul'skiĭab

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk
b Novosibirsk State University, Novosibirsk

Abstract: Many articles deal with large deviation principles (LDPs) (see [1–4] for instance and the references in [3, 4]), studying mainly the LDP for the sums of random elements or for various stochastic models and dynamical systems. For a sequence of random elements in a metric space, in studying LDPs it turns out natural to introduce the concepts of the local LDP and extended LDP. They enable us to state and prove LDP-type statements in those cases when the usual LDP (cf. [3, 4]) fails (see [5, 6] and Section 6 of this article). We obtain conditions for the fulfillment of the extended LDP in metric spaces. The main among these conditions is the fulfillment of the local LDP. The latter is usually much simpler to prove than the extended LDP.

Keywords: large deviation principle, extended large deviation principle, local large deviation principle, deviation function, totally bounded set, compact set.

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English version:
Siberian Mathematical Journal, 2010, 51:6, 989–1003

Bibliographic databases:

Document Type: Article
UDC: 519.21
Received: 01.02.2010

Citation: A. A. Borovkov, A. A. Mogul'skiǐ, “On large deviation principles in metric spaces”, Sibirsk. Mat. Zh., 51:6 (2010), 1251–1269; Siberian Math. J., 51:6 (2010), 989–1003

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Borovkov, “Large deviation principles for random walks with regularly varying distributions of jumps”, Siberian Math. J., 52:3 (2011), 402–410  mathnet  crossref  mathscinet  isi
    2. A. A. Borovkov, A. A. Mogul'skii, “Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories”, Theory Probab. Appl., 56:1 (2012), 21–43  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. A. A. Borovkov, A. A. Mogul'skii, “On large deviation principles for random walk trajectories. I”, Theory Probab. Appl., 56:4 (2011), 538–561  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    4. A. A. Mogul'skiǐ, “The expansion theorem for the deviation integral”, Siberian Adv. Math., 23:4 (2013), 250–262  mathnet  crossref  mathscinet  elib
    5. A. A. Borovkov, A. A. Mogul'skiǐ, “Inequalities and principles of large deviations for the trajectories of processes with independent increments”, Siberian Math. J., 54:2 (2013), 217–226  mathnet  crossref  mathscinet  isi
    6. A. A. Mogul'skiǐ, “On the upper bound in the large deviation principle for sums of random vectors”, Siberian Adv. Math., 24:2 (2014), 140–152  mathnet  crossref  mathscinet  elib
    7. A. A. Borovkov, A. A. Mogul'skiǐ, “Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments”, Siberian Adv. Math., 25:1 (2015), 39–55  mathnet  crossref  mathscinet
    8. A. A. Borovkov, A. A. Mogul'skii, “Moderately large deviation principles for trajectories of random walks and processes with independent increments”, Theory Probab. Appl., 58:4 (2014), 562–581  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    9. A. A. Borovkov, A. A. Mogul'skiǐ, “Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case”, Siberian Adv. Math., 25:4 (2015), 255–267  mathnet  crossref  mathscinet
    10. V. I. Bakhtin, “Spectral potential, Kullback action, and large deviations of empirical measures on measurable spaces”, Theory Probab. Appl., 59:4 (2015), 535–544  mathnet  crossref  crossref  mathscinet  isi  elib
    11. Klebaner F.C., Logachov A.V., Mogulskii A.A., “Large Deviations For Processes on Half-Line”, Electron. Commun. Probab., 20 (2015), 75, 1–14  crossref  mathscinet  isi  scopus
    12. V. I. Bakhtin, “Spektralnyi potentsial, deistvie Kulbaka i printsip bolshikh uklonenii dlya konechno-additivnykh mer”, Tr. In-ta matem., 23:2 (2015), 11–23  mathnet
    13. A. A. Mogul'skiǐ, “The large deviation principle for a compound Poisson process”, Siberian Adv. Math., 27:3 (2017), 160–186  mathnet  crossref  crossref  elib
    14. Bakhtin V., Sokal E., “the Kullback-Leibler Information Function For Infinite Measures”, Entropy, 18:12 (2016), 448  crossref  isi  scopus
    15. Huang G., Mandjes M., Spreij P., “Large Deviations For Markov-Modulated Diffusion Processes With Rapid Switching”, Stoch. Process. Their Appl., 126:6 (2016), 1785–1818  crossref  mathscinet  zmath  isi  scopus
    16. A. A. Mogul'skiǐ, “The extended large deviation principle for a process with independent increments”, Siberian Math. J., 58:3 (2017), 515–524  mathnet  crossref  crossref  isi  elib  elib
    17. N. D. Vvedenskaya, A. V. Logachov, Yu. M. Suhov, A. A. Yambartsev, “A local large deviation principle for inhomogeneous birth-death processes”, Problems Inform. Transmission, 54:3 (2018), 263–280  mathnet  crossref  isi
    18. F. C. Klebaner, A. A. Mogulskii, “Large deviations for processes on half-line: Random Walk and Compound Poisson Process”, Sib. elektron. matem. izv., 16 (2019), 1–20  mathnet  crossref
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