Small deviation probabilities for sums of independent positive random variables
DOI:
https://doi.org/10.21638/spbu01.2020.307Abstract
We examine an asymptotic behavior at zero of distributions and densities of a sum of several independent positive random variables under certain assumptions on the decay rate of their distributions at zero. We consider the cases, when the distributions (densities) of summable random variables are regularly or slowly varying at zero or can decrease at zero with an arbitrary rate.
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Литература
1. Beghin L., Nikitin Ya.Yu., Orsingher E. Exact small ball constants for some Gaussian processes under the L2-norm // Записки научн. сем. ПОМИ. 2003. Т. 298. С. 5–21.
2. Фаталов В.Р. Эргодические средние при большом значении T и точные асимптотики малых уклонений для многомерного винеровского процесса // Изв. РАН. Сер. матем. 2013. Т. 77. Вып. 6. С. 169–206. https://doi.org/10.4213/im7938
3. Никитин Я.Ю., Пусев Р.С. Точная асимптотика малых уклонений для ряда броуновских функционалов // Теория вероятн. и ее примен. 2012. Т. 57. Вып. 1. С. 98–123. https://doi.org/10.4213/tvp4433
4. Розовский Л.В. О вероятностях малых уклонений положительных случайных величин // Записки научн. сем. ПОМИ. 2004. Т. 320. С. 150–159.
5. Розовский Л.В. О сверхбольших уклонениях суммы независимых случайных величин с общим абсолютно непрерывным распределением, удовлетворяющим условию Крамера // Теория вероятн. и ее примен. 2003. Т. 48. Вып. 1. С. 78–103. https://doi.org/10.4213/tvp302
6. Пусев Р. С. Асимптотика малых уклонений процессов Боголюбова в квадратичной норме // Теоретическая и математическая физика. 2010. Т. 165, №1. С. 134–144. https://doi.org/10.4213/tmf6567
7. Никитин Я.Ю., Харинский П.А. Точная асимптотика малых уклонений в L2-норме для одного класса гауссовских процессов // Записки научн. сем. ПОМИ. 2004. Т. 311. С. 214–221.
8. Rozovsky L.V. Comparison theorems for small deviations of weighted series // Probab. math. stat. 2012. Vol. 32, no. 1. P. 117–130.
9. Феллер В. Введение в теорию вероятностей и ее приложения. Т. 2. М.: Мир, 1984.
References
1. Beghin L., Nikitin Ya.Yu., Orsingher E., “Exact small ball constants for some Gaussian processes under the L2-norm”, J.Math. Sci. 128 (1), 2493–2502 (2005). https://doi.org/10.1007/s10958-005-0197-9
2. Fatalov V., “Ergodic means for large values of T and exact asymptotics of small deviations for a multi-dimensional Wiener process”, Izvestiya: Mathematics 77 (6), 1224–1259 (2013). http://dx.doi.org/10.1070/IM2013v077n06ABEH002675
3. Nikitin Ya.Yu., Pusev R. S., “Exact small deviation asymptotics for some Brownian functionals”, Theory Probab. Appl. 57 (1), 60–81 (2013). https://doi.org/10.1137/S0040585X97985790
4. Rozovsky L.V., “On small deviation probabilities of positive random variables”, J. Math. Sci. 137 (1), 4561–4566 (2006). https://doi.org/10.1007/s10958-006-0251-2
5. Rozovsky L.V., “Superlarge Deviations of a Sum of Independent Random Variables Having a Common Absolutely Continuous Distribution under the Cramer Condition”, Theory Probab. Appl. 48 (1), 108–130 (2004). https://doi.org/10.1137/S0040585X980233
6. Pusev R. S., “Asymptotics of small deviations of the Bogoliubov processes with respect to a quadratic norm”, Theor.Math. Phys. 165, 1348–1357 (2010). https://doi.org/10.1007/s11232-010-0113-4
7. Nikitin Ya.Yu., Kharinski P.A., “Sharp small deviation asymptotics in L2-norm for a class of Gaussian processes”, J.Math. Sci. 133 (3), 1328–1332 (2006). https://doi.org/10.1007/s10958-006-0042-9
8. Rozovsky L.V., “Comparison theorems for small deviations of weighted series”, Probab. math. stat. 32 (1), 117–130 (2012).
9. Feller W., Introduction to the probability theory and its applications 2 (John Willey Sons, Inc., 1971).
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Published
2020-09-04
How to Cite
Rozovsky, L. V. (2020). Small deviation probabilities for sums of independent positive random variables. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(3), 435–452. https://doi.org/10.21638/spbu01.2020.307
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
