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 takingvalues 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-dimensionaland X_1 is bounded. The present paper is both an exposition ofLanford's theory and an extension of it to the general case. A numberof examples are considered; these include the cases when X_1 isa Brownian motion or Brownian bridge on the real line, and the casewhen \bar{X}_n is the empirical distribution function based onthe first n values in an i.i.d. sequence of random variables (theSanov problem).