The Annals of Probability

Large Deviations of the Sample Mean in General Vector Spaces

R. R. Bahadur and S. L. Zabell

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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).

Article information

Ann. Probab., Volume 7, Number 4 (1979), 587-621.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F10: Large deviations
Secondary: 62F20 62G20: Asymptotic properties

Random vectors large deviations entropy Sanov's theorem exponential family maximum likelihood


Bahadur, R. R.; Zabell, S. L. Large Deviations of the Sample Mean in General Vector Spaces. Ann. Probab. 7 (1979), no. 4, 587--621. doi:10.1214/aop/1176994985.

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