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).
Publié le : 1979-08-14
Classification:
Random vectors,
large deviations,
entropy,
Sanov's theorem,
exponential family,
maximum likelihood,
60F10,
62F20,
62G20
@article{1176994985,
author = {Bahadur, R. R. and Zabell, S. L.},
title = {Large Deviations of the Sample Mean in General Vector Spaces},
journal = {Ann. Probab.},
volume = {7},
number = {6},
year = {1979},
pages = { 587-621},
language = {en},
url = {http://dml.mathdoc.fr/item/1176994985}
}
Bahadur, R. R.; Zabell, S. L. Large Deviations of the Sample Mean in General Vector Spaces. Ann. Probab., Tome 7 (1979) no. 6, pp. 587-621. http://gdmltest.u-ga.fr/item/1176994985/