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Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments

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Abstract

We extend the large deviation principles for random walks and processes with independent increments to the case of conditional probabilities given that the position of the trajectory at the last time moment is localized in a neighborhood of some point. As a corollary, we obtain a moderately large deviation principle for empirical distributions (an analog of Sanov’s theorem).

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References

  1. A. A. Borovkov, “Boundary-value problems for random walks and large deviations in function spaces,” Teor. Veroyatnost. i Primenen. 12, 635 (1967) [Theory Probab. Appl. 12, 575 (1967)].

    MathSciNet  Google Scholar 

  2. A. A. Borovkov, Probability Theory (Librokom, Moscow, 2009; Springer, London, 2013).

    Google Scholar 

  3. A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks. Vol. I: Slowly Decreasing Jump Distributions. (Fizmatlit, Moscow, 2008) [in Russian].

    Book  Google Scholar 

  4. A. A. Borovkov and A. A. Mogul’skiĭ, “Integro-local and integral theorems for sums of random variables with semiexponential distributions,” Sibirsk. Mat. Zh. 47, 1218 (2006) [Siberian Math. J. 47, 990 (2010)].

    MATH  Google Scholar 

  5. A. A. Borovkov and A. A. Mogul’skiĭ, “On large deviation principles in metric spaces,” Sibirsk. Mat. Zh. 51, 1251 (2010) [Siberian Math. J. 51, 989 (2010)].

    Google Scholar 

  6. A. A. Borovkov and A. A. Mogul’skiĭ, “Chebyshev-type exponential inequalities for sums of random vectors and for trajectories of random walks,” Teor. Veroyatnost. i Primenen. 56, 3 (2011) [Theory Probab. Appl., 56, 21 (2012)].

    Article  Google Scholar 

  7. A. A. Borovkov and A. A. Mogul’skiĭ, “Large deviation principles for random walk trajectories,” I: Teor. Veroyatnost. i Primenen. 56, 627 (2011) [Theory Probab. Appl. 56, 538 (2012)]; II: Teor. Veroyatnost. i Primenen. 57, 3 (2012) [Theory Probab. Appl. 57, 1 (2013)]; III: Teor. Veroyatnost. i Primenen. 58, 37 (2013) [Theory Probab. Appl. 58, 25 (2014)].

    Google Scholar 

  8. A. A. Borovkov and A. A. Mogul’skiĭ, “Moderately large deviation principles for trajectories of random walks and processes with independent increments,” Teor. Veroyatnost. i Primenen. 58, 648 (2013).

    Article  Google Scholar 

  9. I. A. Ibragimov and Yu. V. Linnik, Independent and stationary sequences of random variables (Nauka, Moscow, 1975; Wolters-Noordhoff Publishing, Groningen, 1971).

    Google Scholar 

  10. T. Mikosh and A. V. Nagaev “Large deviations of heavy-tailed sums with applications in insurance,” Extremes 1, 81 (1998).

    Article  MathSciNet  Google Scholar 

  11. A. A. Mogul’skiĭ, “Large deviations for trajectories of multi-dimensional random walks,” Teor. Veroyatnost. i Primenen. 21, 309 (1976) [Theory Probab. Appl. 21, 300 (1977)].

    Google Scholar 

  12. A. V. Nagaev, “Integral limit theorems taking large deviations into account when Cramer’s condition does not hold,” I: Teor. Veroyatnost. i Primenen. 14, 51 (1969) [Theory Probab. Appl. 14, 51 (1969)]; II: Teor. Veroyatnost. i Primenen. 14, 203 (1969) [Theory Probab. Appl. 14, 193 (1969)].

    MathSciNet  Google Scholar 

  13. S. V. Nagaev, “Some limit theorems for large deviations,” Teor. Veroyatnost. i Primenen. 10, 231 (1965). [Theory Probab. Appl. 10, 214 (1965)].

    MathSciNet  Google Scholar 

  14. V. V. Petrov, “Limit theorems for large deviations violating Cramer’s condition. I,” Vestnik Leningrad. Univ. Ser. Mat. Meh. Astronom. No. 19, 49 (1963).

    Google Scholar 

  15. L. V. Rozovskii, “Probabilities of large deviations on the whole axis,” Teor. Veroyatnost. i Primenen. 38, 79 (1993). [Theory Probab. Appl. 38, 53 (1994)].

    MathSciNet  Google Scholar 

  16. L. Saulis and V. Staulevečius, Limit Theorems for Large Deviations (Mokslas, Vilnius, 1989; Kluwer, Dordrecht, 1991).

    MATH  Google Scholar 

  17. C. Stone, “On local and ratio limit theorems,” Proc. Fifth Berkeley Sympos. Math. Statist. and Probability. Berkeley (Berkeley, Calif., 1965/66), V. II: Contributions to Probability Theory, Part 2, 217 (Univ. California Press, Berkeley, Calif., 1965/1966).

    Google Scholar 

  18. S. R. S. Varadhan, “Asymptotic probabilities and differential equations,” Comm. Pure Appl. Math. 19, 261 (1966).

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, 630090, Russia

    A. A. Borovkov & A. A. Mogul’skiĭ

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    A. A. Borovkov & A. A. Mogul’skiĭ

Authors
  1. A. A. Borovkov
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  2. A. A. Mogul’skiĭ
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Correspondence to A. A. Borovkov.

Additional information

Original Russian Text © A. A. Borovkov, A. A. Mogul’skiĭ, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 2, pp. 45–68.

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Borovkov, A.A., Mogul’skiĭ, A.A. Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments. Sib. Adv. Math. 25, 39–55 (2015). https://doi.org/10.3103/S1055134415010058

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