NDLI: On Large and Superlarge Deviations of Sums of Independent Random Vectors Under Cramr's Condition. II

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Borovkov, A. A. Mogulskii, A. A.
Copyright Year	2007
Abstract	The present paper is a continuation of [A. A. Borovkov and A. A. Mogulskii, Theory Probab. Appl., 51 (2007), pp. 227255]. 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 Cramr transforms with parameter values close to the critical" one. Moreover, we obtain asymptotic expansion for the large deviations rate function.
Starting Page	567
Ending Page	594
Page Count	28
File Format	PDF
ISSN	0040585X
DOI	10.1137/S0040585X97982645
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	4
Volume Number	51
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2007-11-19
Access Restriction	Subscribed
Subject Keyword	large deviations superlarge deviations integrolocal theorem characterization of the normal distribution asymptotic expansions of the large deviations rate function semi-exponential distributions superexponential distributions limit theorems for Cramr transforms large deviations rate function
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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