Let X1, X2, ··· be a sequence of i.i.d. random vectors taking values in a space V, let X-n = (X1 + ··· + Xn)/n, and for J ⊂ V let an(J) = n-1log P(X-n∈ J). A powerful theory concerning the existence and value of limn→∞ an(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)
