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August, 1979 Large Deviations of the Sample Mean in General Vector Spaces
R. R. Bahadur, S. L. Zabell
Ann. Probab. 7(4): 587-621 (August, 1979). DOI: 10.1214/aop/1176994985


Let $X_1, X_2, \cdots$ be a sequence of i.i.d. random vectors taking values in a space $V$, let $\bar{X}_n = (X_1 + \cdots + X_n)/n$, and for $J \subset V$ let $a_n(J) = n^{-1} \log P(\bar{X}_n \in J)$. A powerful theory concerning the existence and value of $\lim_{n\rightarrow\infty} a_n(J)$ has been developed by Lanford for the case when $V$ is finite-dimensional and $X_1$ is bounded. The present paper is both an exposition of Lanford's theory and an extension of it to the general case. A number of examples are considered; these include the cases when $X_1$ is a Brownian motion or Brownian bridge on the real line, and the case when $\bar{X}_n$ is the empirical distribution function based on the first $n$ values in an i.i.d. sequence of random variables (the Sanov problem).


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R. R. Bahadur. S. L. Zabell. "Large Deviations of the Sample Mean in General Vector Spaces." Ann. Probab. 7 (4) 587 - 621, August, 1979.


Published: August, 1979
First available in Project Euclid: 19 April 2007

zbMATH: 0424.60028
MathSciNet: MR537209
Digital Object Identifier: 10.1214/aop/1176994985

Primary: 60F10
Secondary: 62F20 , 62G20

Keywords: Entropy , exponential family , large deviations , maximum likelihood , random vectors , Sanov's theorem

Rights: Copyright © 1979 Institute of Mathematical Statistics


Vol.7 • No. 4 • August, 1979
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