Multivariate Probabilities of Large Deviations

Abstract

Let ${P_n}^{\infty_n=1}$ denote a sequence of probability measures on $(R^k,B^k)$, where $R^k$ is k-dimensional Euclidean space and $B^k$ the Borel subsets. For $A\inB^k$, let $e(A)=lim_{n\rigtharrow\infty}(1/n) log P_n(A)$, if the limit exists. Sufficient conditions are given for expressing e(A) as the supremum of e(B) for certain "rectangular" sets $B=X^k_{j=1}(\alphaj,\betaj)$ with either $\alphaj=-\infty or \betaj=+\infty$ for each j = 1, ..., k. Also, some k-dimensional generalizations of the density theorem of Killeen, et al. (1972) are given for expressing e(A) in terms of certain limits of the sequence of density (or probability) functions. Finally, an example is considered where $P_n$ is the distribution of k order statistics from a sample of size n.