On probabilities of small deviations for stochastic processes. (English. Russian original) Zbl 0718.60024
Sib. Adv. Math. 1, No. 1, 39-63 (1991); translation from Tr. Inst. Mat. 13, 147-168 (1989).
See the review in Zbl 0708.60029.
MSC:
| 60F10 | Large deviations |
| 60F17 | Functional limit theorems; invariance principles |
| 60G15 | Gaussian processes |
| 60J65 | Brownian motion |
References:
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| [2] | A. A. Borovkov, Probability Theory (Librokom, Moscow, 2009; Springer, London, 2013). |
| [3] | A. A. Borovkov and K. A. Borovkov, Asymptotic Analysis of Random Walks. Vol. I: Slowly Decreasing Jump Distributions. (Fizmatlit, Moscow, 2008) [in Russian]. · Zbl 1231.60001 |
| [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)]. · Zbl 1150.60021 |
| [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)]. · Zbl 1225.60051 |
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| [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). · Zbl 1340.60020 · doi:10.4213/tvp4534 |
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| [10] | T. Mikosh and A. V. Nagaev ”Large deviations of heavy-tailed sums with applications in insurance,” Extremes 1, 81 (1998). · Zbl 0927.60037 · doi:10.1023/A:1009913901219 |
| [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)]. |
| [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)]. · Zbl 0181.45004 |
| [13] | S. V. Nagaev, ”Some limit theorems for large deviations,” Teor. Veroyatnost. i Primenen. 10, 231 (1965). [Theory Probab. Appl. 10, 214 (1965)]. · Zbl 0144.18704 |
| [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). |
| [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)]. |
| [16] | L. Saulis and V. Staulevečius, Limit Theorems for Large Deviations (Mokslas, Vilnius, 1989; Kluwer, Dordrecht, 1991). · Zbl 0714.60018 |
| [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). |
| [18] | S. R. S. Varadhan, ”Asymptotic probabilities and differential equations,” Comm. Pure Appl. Math. 19, 261 (1966). · Zbl 0147.15503 · doi:10.1002/cpa.3160190303 |
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