Geodesic
Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories
Teoriâ veroâtnostej i ee primeneniâ, Tome 56 (2011) no. 1, pp. 3-29 Cet article a éte moissonné depuis la source Math-Net.Ru

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We obtain analogues of the well-known Chebyshev's exponential inequality $\mathbf P(\xi \ge x)\le e^{-\Lambda^{(\xi)}(x)}$, $x>\mathbf E\,\xi$, for the distribution of a random variable $\xi$, where $\Lambda^{(\xi)}(x):=\sup_\lambda\{\lambda x- \log \mathbf E\,e^{\lambda \xi}\}$ is the large deviation rate function for $\xi$. Generalizations of this relation are established for multivariate random vectors $\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums.
Mots-clés : Cramér condition
Keywords: large deviation rate function, random walk, deviation functional, action functional, convex set, large deviations, large deviation principle, extended large deviation principle, inequalities for large deviations. 
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A. A. Borovkov; A. A. Mogul'skii. Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories. Teoriâ veroâtnostej i ee primeneniâ, Tome 56 (2011) no. 1, pp. 3-29. http://geodesic.mathdoc.fr/item/TVP_2011_56_1_a0/

[1] Borovkov A. A., Mogulskii A. A., “O printsipakh bolshikh uklonenii v metricheskikh prostranstvakh”, Cib. matem. zhurn., 51:6 (2010), 1251–1269 | MR | Zbl

[2] Borovkov A. A., Mogulskii A. A., Rasshirennyi printsip bolshikh uklonenii dlya traektorii sluchainykh bluzhdanii. I, II, Preprint, IM SO RAN, Novosibirsk, 2010

[3] Borovkov A. A., Teoriya veroyatnostei, Editorial URSS, M., 2009, 652 pp.

[4] Borovkov A. A., Mogulskii A. A., Bolshie ukloneniya i proverka statisticheskikh gipotez, Nauka, Novosibirsk, 1992, 222 pp.

[5] Funktsionalnyi analiz, Nauka, M., 1964, 424 pp. (Spravochnaya matematicheskaya biblioteka)

[6] Rokafellar R., Vypuklyi analiz, Mir, M., 1973, 469 pp.

[7] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1968, 496 pp.

[8] Borovkov A. A., “Nekotorye neravenstva dlya summ mnogomernykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 13:1 (1968), 155–159 | MR | Zbl

[9] Borovkov A. A., “Granichnye zadachi dlya sluchainykh bluzhdanii i bolshie ukloneniya v funktsionalnykh prostranstvakh”, Teoriya veroyatn. i ee primen., 12:4 (1967), 635–654 | MR | Zbl

[10] Mogulskii A. A., “Bolshie ukloneniya dlya traektorii mnogomernykh sluchainykh bluzhdanii”, Teoriya veroyatn. i ee primen., 21:2 (1976), 309–323 | MR | Zbl

[11] Varadhan S. R. S., “Asymptotic probabilities and differential equations”, Comm. Pure Appl. Math., 19:3 (1966), 261–286 | DOI | MR | Zbl

[12] Borovkov A. A., Mogulskii A. A., Svoistva funktsionala uklonenii ot traektorii, voznikayuschego pri analize veroyatnostei bolshikh uklonenii sluchainykh bluzhdanii, Preprint, IM SO RAN, Novosibirsk, 2010

[13] Borovkov A. A., Mogulskii A. A., “O bolshikh i sverkhbolshikh ukloneniyakh summ nezavisimykh sluchainykh vektorov pri vypolnenii usloviya Kramera. I, II”, Teoriya veroyatn. i ee primen., 51:2 (2006), 260–294 ; 4, 641–673