Abstract:
We obtain analogues of the well-known Chebyshev's exponential inequality
P(ξ⩾x)⩽e−Λ(ξ)(x), x>Eξ, for the distribution of a random variable ξ, where Λ(ξ)(x):=supλ{λx−logEeλξ} is the large deviation rate function for ξ. Generalizations of this relation are established for multivariate random vectors ξ, for sums of the vectors, and for trajectories of random processes associated with such sums.
Keywords:
Cramér condition, 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.
Citation:
A. A. Borovkov, A. A. Mogul'skii, “Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories”, Teor. Veroyatnost. i Primenen., 56:1 (2011), 3–29; Theory Probab. Appl., 56:1 (2012), 21–43
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\paper Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories
\jour Teor. Veroyatnost. i Primenen.
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\pages 3--29
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\jour Theory Probab. Appl.
\yr 2012
\vol 56
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\pages 21--43
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Linking options:
https://www.mathnet.ru/eng/tvp4321
https://doi.org/10.4213/tvp4321
https://www.mathnet.ru/eng/tvp/v56/i1/p3
This publication is cited in the following 20 articles:
Ryan T. White, “On the Exiting Patterns of Multivariate Renewal-Reward Processes with an Application to Stochastic Networks”, Symmetry, 14:6 (2022), 1167
A. V. Logachov, A. A. Mogul'skii, “Exponential chebyshev inequalities for random graphons and their applications”, Siberian Math. J., 61:4 (2020), 697–714
A. V. Logachev, A. A. Mogulskii, “Lokalnye teoremy dlya konechnomernykh priraschenii arifmeticheskikh mnogomernykh obobschennykh protsessov vosstanovleniya pri vypolnenii usloviya Kramera”, Sib. elektron. matem. izv., 17 (2020), 1766–1786
A. A. Mogulskii, “Lokalnye teoremy dlya arifmeticheskikh obobschennykh protsessov vosstanovleniya pri vypolnenii usloviya Kramera”, Sib. elektron. matem. izv., 16 (2019), 21–41
A. A. Mogul'skiǐ, E. I. Prokopenko, “Local theorems for arithmetic multidimensional compound renewal processes under Cramér's condition”, Siberian Adv. Math., 30:4 (2020), 284–302
Dolera E., Regazzini E., “Uniform Rates of the Glivenko-Cantelli Convergence and Their Use in Approximating Bayesian Inferences”, Bernoulli, 25:4A (2019), 2982–3015
A. A. Mogul'skiǐ, “The extended large deviation principle for a process with independent increments”, Siberian Math. J., 58:3 (2017), 515–524
A. A. Mogul'skiǐ, “The large deviation principle for a compound Poisson process”, Siberian Adv. Math., 27:3 (2017), 160–186
A. A. Borovkov, A. A. Mogul'skiǐ, “Large deviation principles for sums of random vectors and the corresponding renewal functions in the inhomogeneous case”, Siberian Adv. Math., 25:4 (2015), 255–267
A. A. Borovkov, A. A. Mogul'skiǐ, “Inequalities and principles of large deviations for the trajectories of processes with independent increments”, Siberian Math. J., 54:2 (2013), 217–226
A. A. Mogul'skiǐ, “On the upper bound in the large deviation principle for sums of random vectors”, Siberian Adv. Math., 24:2 (2014), 140–152
A. A. Borovkov, A. A. Mogulskii, “Large deviation principles for random walk trajectories. III”, Theory Probab. Appl., 58:1 (2014), 25–37
A. A. Borovkov, A. A. Mogul'skiǐ, “Conditional moderately large deviation principles for the trajectories of random walks and processes with independent increments”, Siberian Adv. Math., 25:1 (2015), 39–55
A. A. Borovkov, A. A. Mogul'skii, “Moderately large deviation principles for trajectories of random walks and processes with independent increments”, Theory Probab. Appl., 58:4 (2014), 562–581
Aurelija Kasparavičiūtė, Theorems of Large Deviations for the Sums of a Random Number of Independent Random Variables, 2013
A. A. Borovkov, A. A. Mogul'skii, “On large deviation principles for random walk trajectories. II”, Theory Probab. Appl., 57:1 (2013), 1–27
A. A. Mogul'skiǐ, “The expansion theorem for the deviation integral”, Siberian Adv. Math., 23:4 (2013), 250–262
A. A. Borovkov, A. A. Mogul'skiǐ, “Properties of a functional of trajectories which arises in studying the probabilities of large deviations of random walks”, Siberian Math. J., 52:4 (2011), 612–627
A. A. Borovkov, A. A. Mogul'skii, “On large deviation principles for random walk trajectories. I”, Theory Probab. Appl., 56:4 (2011), 538–561
A. A. Borovkov, A. A. Mogul'skiǐ, “On large deviation principles in metric spaces”, Siberian Math. J., 51:6 (2010), 989–1003
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