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 Teor. Veroyatnost. i Primenen., 2006, Volume 51, Issue 2, Pages 260–294 (Mi tvp53)

On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I

A. A. Borovkov, A. A. Mogul'skii

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

Abstract: We study the asymptotics of the probability that the sum of independent identically distributed random vectors is in a small cube with a vertex at point $x$ in the following two problems. (A) When the relative (normalized) deviations $x/n$ ($n$ is the number of terms in the sum) are in the analyticity domain of the large deviation rate function $\Lambda(\alpha)$ for the summands (if, in addition, $|x|/n\to\infty$, then one speaks of super-large deviations). (B) When the alternative possibility takes place, i.e., when $x/n$ is outside the analyticity domain of the function $\Lambda(\alpha)$. In problems (A) and (B) the asymptotics of the super-large deviation probabilities (when $|x/n|\to\infty$), just as the asymptotics of the probabilities of the “usual” large deviation in problem (B) (when $x/n$ is bounded away from the expectation of the summands and remains bounded), in many aspects remained unknown. The present paper, consisting of two parts, is mostly devoted to solving problem (A) for super-large deviations. In part I we present a solution to problem (A) in the general multivariate case. As the first step, we use the Cramér transform, which enables one to reduce the problem on super-large deviations of the original sum to that on normal deviations of the sum of the transformed random vectors. Then we use integrolocal or local theorems for sums of random vectors in the triangular array scheme in the normal deviations zone. The required versions of such theorems are contained in [A. A. Borovkov and A. A. Mogulskii, Math. Notes, 79 (2006), pp. 468–482] and in section 5. We also present in part I a scheme for solving problem (B), to which a separate paper will be devoted. In the case when the distribution of the sum is absolutely continuous in a neighborhood of the point $x$, we study the asymptotics of the respective density at that point.

Keywords: rate function, large deviations, super-large deviations, integrolocal theorem, triangular array scheme, Cramér transform.

DOI: https://doi.org/10.4213/tvp53

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English version:
Theory of Probability and its Applications, 2007, 51:2, 227–255

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Citation: A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I”, Teor. Veroyatnost. i Primenen., 51:2 (2006), 260–294; Theory Probab. Appl., 51:2 (2007), 227–255

Citation in format AMSBIB
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This publication is cited in the following articles:
1. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations of sums of independent random vectors under Cramér's condition. II”, Theory Probab. Appl., 51:4 (2007), 567–594
2. 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
3. L. V. Rozovskii, “Superlarge deviation probabilities for sums of independent random variables with exponential decreasing distribution”, Theory Probab. Appl., 52:1 (2008), 167–171
4. 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
5. 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
6. 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
7. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271
8. A. A. Borovkov, A. A. Mogul'skiǐ, “On large deviation principles in metric spaces”, Siberian Math. J., 51:6 (2010), 989–1003
9. 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
10. Peter Eichelsbacher, Thomas Kriecherbauer, Katharina Schüler, “Precise Deviations Results for the Maxima of Some Determinantal Point Processes: the Upper Tail”, SIGMA, 12 (2016), 093, 18 pp.
11. G. A. Bakai, A. V. Shklyaev, “Large deviations of generalized renewal process”, Discrete Math. Appl., 30:4 (2020), 215–241
12. Trojan B., “Long Time Behavior of Random Walks on the Integer Lattice”, Mon.heft. Math., 191:2 (2020), 349–376
13. G. A. Bakai, “Bolshie ukloneniya dlya obryvayuschegosya obobschennogo protsessa vosstanovleniya”, Teoriya veroyatn. i ee primen., 66:2 (2021), 261–283
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