Large deviations of the sample mean in general vector spaces

Abstract

Let X₁, X₂, ··· be a sequence of i.i.d. random vectors taking values in a space V, let X^-_n = (X₁ + ··· + X_n)/n, and for J ⊂ V let a_n(J) = n^-1log P(X^-_n∈ J). A powerful theory concerning the existence and value of lim_n→∞ a_n(J) has been developed by Lanford for the case when V is finite-dimensional and X₁ 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 X₁ is a Brownian motion or Brownian bridge on the real line, and the case when 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).