Abstract
We consider large deviation principles (LDPs) for random walks and renewal compound processes. We expose and analyze the approaches to finding existence conditions for “exact” large deviation principles valid under minimal moment conditions. By exact large deviation principles we understand statements on the existence of exact limits for the asymptotics of the probabilities under study and the values of these limits rather than on the existence of upper and lower limits for these asymptotics, as for the ordinary large deviation principles. In addition to the known results, the new versions of conditions are found that ensure the existence of exact LDPs under the minimal Cramér’s moment condition. The article is based on the text of a talk prepared for a conference cancelled because of the pandemic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
In Definition 1 of [1] the condition of continuity of \( I(f) \) at the points of \( \partial B_{} \) with respect to the distance \( \rho_{𝔻}(f^{\prime},g^{\prime}) \) is omitted.
References
Borovkov A. A., “On exact large deviation principles for compound renewal processes,” Theory Probab. Appl., vol. 66, no. 2, 170–183 (2021).
Borovkov A. A., Logachev A. V., and Mogulskii A. A., “Chebyshev-type inequalities and large deviation principles,” Theory Probab. Appl., vol. 66, no. 4, 718–733 (2021).
Borovkov A. A., Asymptotic Analysis of Random Walks: Light-Tailed Distributions, Cambridge University, Cambridge (2020).
Borovkov A. A., Compound Renewal Processes, Russian Academy of Sciences, Moscow (2020) [Russian] (to appear in Cambridge University).
Borovkov A. A., “Boundary problems for random walks and large deviations in function spaces,” Theory Probab. Appl., vol. 12, no. 4, 575–595 (1967).
Puhalskii A. A., “On functional principle of large deviations,” in: New Trends in Probability and Statistics. I, VSP/Mokslas, Utrecht and Vilnius (1991), 198–219.
Borovkov A. A. and Mogulskii A. A., “Large deviation principles for random walk trajectories. I,” Theory Probab. Appl., vol. 56, no. 4, 538–561 (2012); “Large deviation principles for random walk trajectories. II,” vol. 57, no. 1, 1–27 (2013); “Large deviation principles for random walk trajectories. III,” vol. 58, no. 1, 25–37 (2014).
Skorokhod A. V., “Limit theorems for stochastic processes,” Theory Probab. Appl., vol. 1, no. 3, 261–290 (1956).
Mogulskii A. A., “The extended principle of large deviations for compound renewal processes of trajectories,” Mat. Tr., vol. 24, no. 1, 142–174 (2021).
Dembo A. and Zeitouni O., Large Deviations Techniques and Applications. 2nd ed., Springer, New York (1998).
Feng J. and Kurtz T. G., Large Deviations for Stochastic Processes. Math. Surv. Monogr.; Vol. 131, Amer. Math. Soc., Providence (2006).
Acknowledgment
The author is grateful to A.A. Mogulskii and A.V. Logachev for useful discussions.
Funding
The author was supported by the Program of Basic Scientific Research of the Siberian Branch of the Russian Academy of Sciences (Grant no. I.1.3, Project 0314–2016–0008) as well as funded by the Russian Foundation for Basic Research (Grant no. 18–01–00101a).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 1, pp. 58–76. https://doi.org/10.33048/smzh.2022.63.104
Rights and permissions
About this article
Cite this article
Borovkov, A.A. On the Existence Conditions for Exact Large Deviation Principles. Sib Math J 63, 48–64 (2022). https://doi.org/10.1134/S0037446622010049
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622010049
Keywords
- random walks
- renewal compound processes
- large deviation principle
- extended large deviation principle
- exact large deviation principle
- thin boundaries