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Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks

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Abstract

The deviation functional (or integral) describes the logarithmic asymptotics of the probabilities of large deviations of trajectories of the random walks generated by the sums of random variables (vectors) (see [1, 2] for instance). In this article we define it on a broader function space than previously and under weaker assumptions on the distributions of jumps of the random walk. The deviation integral turns out the Darboux integral ∫ F(t, u) of a semiadditive interval function F(t, u) of a particular form. We study the properties of the deviation integral and use the results elsewhere in [3] to prove some generalizations of the large deviation principle established previously under rather restrictive assumptions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics and Novosibirsk State University, Novosibirsk, 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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Corresponding author

Correspondence to A. A. Borovkov.

Additional information

Original Russian Text Copyright © 2011 Borovkov A. A. and Mogul’skiĭ A. A.

The authors were supported by the Russian Foundation for Basic Research (Grants 05-01-00810, 07-01-00595, and 08-01-00962) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grants NSh-8980.2006.1 and RNSh.2.1.1.1379).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 52, No. 4, pp. 777–795, July–August, 2011.

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Borovkov, A.A., Mogul’skiĭ, A.A. Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks. Sib Math J 52, 612–627 (2011). https://doi.org/10.1134/S0037446611040069

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  • DOI: https://doi.org/10.1134/S0037446611040069

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