Large Deviations of the Sample Mean in General Vector Spaces

Abstract

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).