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 Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 4, Pages 641–673 (Mi tvp18)

On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II

A. A. Borovkov, A. A. Mogul'skii

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227–255]. In that paper we studied, in the univariate case, the asymptotics of the probabilities that a sum of independent identically distributed random variables will hit a half-interval $[x,x+\Delta)$ in the zone of superlarge deviations when the relative (scaled) deviations $\alpha=x/n$ unboundedly increase together with the number of summands $n$ and, at the same time, remain in the analyticity domain of the large deviations rate function for the summands. In the multivariate case, the first part of the paper presented sufficient conditions which ensure that integrolocal and local theorems of the same universal type as in the large and normal deviations zones will also hold in the superlarge deviations zone. The second part of the paper deals with the same problems for three classes on the most wide-spread univariate distributions, for which one can obtain simple sufficient conditions, enabling one to find the asymptotics of the desired probabilities, as $x/n\to \infty$, in the above-mentioned universal form. These are the classes of the so-called exponentially and “superexponentially” fast decaying regular distributions. For them, we also establish limit theorems for the Cramér transforms with parameter values close to the “critical” one. Moreover, we obtain asymptotic expansion for the large deviations rate function.

Keywords: large deviations rate function, large deviations, superlarge deviations, integrolocal theorem, semi-exponential distributions, superexponential distributions, characterization of the normal distribution, limit theorems for Cramér transforms, asymptotic expansions of the large deviations rate function.

DOI: https://doi.org/10.4213/tvp18

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English version:
Theory of Probability and its Applications, 2007, 51:4, 567–594

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Citation: A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II”, Teor. Veroyatnost. i Primenen., 51:4 (2006), 641–673; Theory Probab. Appl., 51:4 (2007), 567–594

Citation in format AMSBIB
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• https://doi.org/10.4213/tvp18
• http://mi.mathnet.ru/eng/tvp/v51/i4/p641

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This publication is cited in the following articles:
1. A. A. Borovkov, A. A. Mogul'skii, “Integro-local and integral theorems for sums of random variables with semiexponential distributions”, Siberian Math. J., 47:6 (2006), 990–1026
2. A. A. Mogulskiǐ, Ch. Pagma, “Superlarge deviations for sums of random variables with arithmetical super-exponential distributions”, Siberian Adv. Math., 18:3 (2008), 185–208
3. A. A. Mogul'skii, “An integro-local theorem applicable on the whole half-axis to the sums of random variables with regularly varying distributions”, Siberian Math. J., 49:4 (2008), 669–683
4. A. A. Borovkov, “Tauberian and Abelian theorems for rapidly decaying distributions and their applications to stable laws”, Siberian Math. J., 49:5 (2008), 796–805
5. A. A. Borovkov, A. A. Mogul'skii, “On Large Deviations of Sums of Independent Random Vectors on the Boundary and Outside of the Cramér Zone. I”, Theory Probab. Appl., 53:2 (2009), 301–311
6. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271
7. A. A. Borovkov, A. A. Mogul'skiǐ, “On large deviation principles in metric spaces”, Siberian Math. J., 51:6 (2010), 989–1003
8. A. A. Borovkov, A. A. Mogul'skii, “Chebyshev type exponential inequalities for sums of random vectors and random walk trajectories”, Theory Probab. Appl., 56:1 (2012), 21–43
9. Rozovsky L., “Super large deviation probabilities for sums of independent lattice random variables with exponential decreasing tails”, Statistics & Probability Letters, 82:1 (2012), 72–76
10. N. V. Gribkova, R. Helmers, “Second order approximations for slightly trimmed means”, Theory Probab. Appl., 58:3 (2014), 383–412
11. L. V. Rozovskii, “Superlarge deviation probabilities for sums of independent random variables with exponential decreasing distributions. II”, Theory Probab. Appl., 59:1 (2015), 168–177
12. Fan X., “Sharp Large Deviations For Sums of Bounded From Above Random Variables”, Sci. China-Math., 60:12 (2017), 2465–2480
13. L. V. Rozovskii, “Ob asimptotike svertki raspredelenii s regulyarno eksponentsialno ubyvayuschimi khvostami”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 265–274
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