Abstract
We establish an extended large deviation principle for processes with independent and stationary increments on the half-line under the Cramer moment condition in the space of functions of bounded variation without discontinuities of the second kind equipped with the Borovkov metric.
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References
Borovkov, A.A., Teoriya veroyatnostei (Probability Theory), Moscow: Librokom, 2009, 5th ed.
Borovkov, A.A., The Convergence of Distributions of Functionals of Stochastic Processes, Uspekhi Mat. Nauk, 1972, vol. 27, no. 1 (163), pp. 3–41 [Russian Math. Surveys (Engl. Transl.), 1972, vol. 27, no. 1, pp. 1–42].
Borovkov, A.A., The Rate of Convergence in the Invariance Principle, Teor. Veroyatn. Primen., 1973, vol. 18, no. 2, pp. 217–234 [Theory Probab. Appl. (Engl. Transl.), 1974, vol. 18, no. 2, pp. 207–225].
Skorokhod, A.V., Limit Theorems for Random Processes, Teor. Veroyatnost. i Primenen., 1956, vol. 1, no. 2, pp. 289–319 [Theory Probab. Appl. (Engl. Transl.), 1956, vol. 1, no. 3, pp. 261–290].
Borovkov, A.A. and Mogulskii, A.A., Large Deviation Principles for Random Walk Trajectories. I, II, III, Teor. Veroyatn. Primen., 2011, vol. 56, no. 4, pp. 627–655; 2012, vol. 57, no. 1, pp. 3–34; 2013, vol. 58, no. 1, pp. 37–52 [Theory Probab. Appl. (Engl. Transl.), 2012, vol. 56, no. 4, pp. 538–561; 2013, vol. 57, no. 1, pp. 1–27; 2014, vol. 58, no. 1, pp. 25–37].
Mogul’skiĭ, A.A., The Large Deviation Principle for a Compound Poisson Process, Mat. Tr., 2016, vol. 19, no. 2, pp. 119–157 [Siberian Adv. Math. (Engl. Transl.), 2017, vol. 27, no. 3, pp. 160–186].
Borovkov, A.A., Asimptoticheskii analiz sluchainykh bluzhdanii. Bystro ubyvayushchie raspredeleniya prirashchenii (Asymptotic Analysis of Random Walks. Rapidly Decreasing Distributions of Increments), Moscow: Fizmatlit, 2013.
Klebaner, F.C., Logachov, A.V., and Mogulskii, A.A., Large Deviations for Processes on Half-line, Electron. Commun. Probab., 2015, vol. 20, Paper no. 75 (14 pp.).
Riesz, F. and Szőkefalvi-Nagy, B., Leçons d’analyse fonctionnelle, Budapest: Akad. Kiadó, 1952. Translated under the title Lektsii po funktsional’nomu analizu, Moscow: Inostr. Lit., 1953.
Borovkov, A.A., Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces, Teor. Veroyatnost. i Primenen., 1967, vol. 12, no. 4, pp. 635–654 [Theory Probab. Appl. (Engl. Transl.), 1967, vol. 12, no. 4, pp. 575–595].
Borovkov, A.A. and Mogulskii, A.A., Bol’shie ukloneniya i proverka statisticheskikh gipotez (Large Deviations and Statistical Hypotheses Testing), Novosibirsk: Nauka, 1992.
Borovkov, A.A. and Mogul’skiĭ, A.A., Properties of a Functional of Trajectories Which Arises in Studying the Probabilities of Large Deviations of Random Walks, Sibirsk. Mat. Zh., 2011, vol. 52, no. 4, pp. 777–795 [Sib. Math. J. (Engl. Transl.), 2011, vol. 52, no. 4, pp. 612–627].
Mogul’skiĭ, A.A., The Expansion Theorem for the Deviation Integral, Mat. Tr., 2012, vol. 15, no. 2, pp. 127–145 [Siberian Adv. Math. (Engl. Transl.), 2013, vol. 23, no. 4, pp. 250–262].
Klebaner F.C., Mogulskii A.A. Large Deviations for Processes on Half-line: Random Walk and Compound Poisson Process, Siberian Electron. Math. Rep., 2019, vol. 16, pp. 1–20. Available at http://semr.math.nsc.ru/v16/p1-20.pdf.
Borovkov, A.A. and Mogul’skiĭ, A.A., Inequalities and Principles of Large Deviations for the Trajectories of Processes with Independent Increments, Sibirsk. Mat. Zh., 2013, vol. 54, no. 2, pp. 286–297 [Sib. Math. J. (Engl. Transl.), 2013, vol. 54, no. 2, pp. 217–226].
Borovkov, A.A. and Mogulskii, A.A., Moderately Large Deviation Principles for the Trajectories of Random Walks and Processes with Independent Increments, Teor. Veroyatn. Primen., 2013, vol. 58, no. 4, pp. 648–671 [Theory Probab. Appl. (Engl. Transl.), 2014, vol. 58, no. 4, pp. 562–581].
Lynch, J. and Sethuraman, J., Large Deviations for Processes with Independent Increments, Ann. Probab., 1987, vol. 15, no. 2, pp. 610–627.
Mogul’skiĭ, A.A., The Extended Large Deviation Principle for a Process with Independent Increments, Sibirsk. Mat. Zh., 2017, vol. 58, no. 3, pp. 660–672 [Sib. Math. J. (Engl. Transl.), 2017, vol. 58, no. 3, pp. 515–524].
Dobrushin, R.L. and Pechersky, E.A., Large Deviations for Random Processes with Independent Increments on Infinite Intervals, Probl. Peredachi Inf., 1998, vol. 34, no. 4, pp. 76–108 [Probl. Inf. Transm. (Engl. Transl.), 1998, vol. 34, no. 4, pp. 354–382].
Borovkov, A.A. and Mogul’skii, A.A., On Large Deviation Principles in Metric Spaces, Sibirsk. Mat. Zh., 2010, vol. 51, no. 6, pp. 1251–1269 [Sib. Math. J. (Engl. Transl.), 2010, vol. 51, no. 6, pp. 989–1003].
Doob, J.L., Stochastic Processes, New York: Wiley, 1953. Translated under the title Veroyatnostnye protsessy, Moscow: Inostr. Lit., 1956.
Acknowledgements
The authors are grateful to E.A. Pechersky for useful discussions and very valuable comments.
Funding
The research of F.C. Klebaner was supported in part by the Australian Research Council grant DP150102758. The research of A.V. Logachov was supported in part by the Program of Fundamental Scientific Research of the Siberian Branch of the Russian Academy of Sciences no. 1.1.3, project no. 0314-2016-0008. The research of A.A. Mogulskii was supported in part by the Russian Foundation for Basic Research, project no. 18-01-00101.
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Russian Text © The Author(s), 2020, published in Problemy Peredachi Informatsii, 2020, Vol. 56, No. 1, pp. 63–79.
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Klebaner, F.C., Logachov, A.V. & Mogulskii, A.A. Extended Large Deviation Principle for Trajectories of Processes with Independent and Stationary Increments on the Half-line. Probl Inf Transm 56, 56–72 (2020). https://doi.org/10.1134/S0032946020010068
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DOI: https://doi.org/10.1134/S0032946020010068