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July, 1975 Multivariate Probabilities of Large Deviations
Gerald L. Sievers
Ann. Statist. 3(4): 897-905 (July, 1975). DOI: 10.1214/aos/1176343190


Let ${P_n}^{\infty_n=1}$ denote a sequence of probability measures on $(R^k,B^k)$, where $R^k$ is k-dimensional Euclidean space and $B^k$ the Borel subsets. For $A\inB^k$, let $e(A)=lim_{n\rigtharrow\infty}(1/n) log P_n(A)$, if the limit exists. Sufficient conditions are given for expressing e(A) as the supremum of e(B) for certain "rectangular" sets $B=X^k_{j=1}(\alphaj,\betaj)$ with either $\alphaj=-\infty or \betaj=+\infty$ for each j = 1, ..., k. Also, some k-dimensional generalizations of the density theorem of Killeen, et al. (1972) are given for expressing e(A) in terms of certain limits of the sequence of density (or probability) functions. Finally, an example is considered where $P_n$ is the distribution of k order statistics from a sample of size n.


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Gerald L. Sievers. "Multivariate Probabilities of Large Deviations." Ann. Statist. 3 (4) 897 - 905, July, 1975.


Published: July, 1975
First available in Project Euclid: 12 April 2007

zbMATH: 0313.60023
MathSciNet: MR375622
Digital Object Identifier: 10.1214/aos/1176343190

Primary: 60F10
Secondary: 62G20 , 62G30

Keywords: density limit , exact slope , large deviations , order statistics

Rights: Copyright © 1975 Institute of Mathematical Statistics

Vol.3 • No. 4 • July, 1975
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