On Cramér’s Theorems Concerning Weak Convergence of Distributions

Abstract

In many statistical investigations, three theorems of Cram6r [1, 20.6, p. 254] are being applied which deal with the influence of "small changes in probability" upon the asymptotic distribution of sequences of random variables. The purpose of this note is to show that they are particular cases of a single theorem which, in addition, refers to distributions in spaces of arbitrary dimension, and whose proof is at least as easy as that of each of the three original theorems. By Q, we denote the probability distribution of a random vector x which takes values in the real space R h of k dimensions; thus Q, is a probability measure in R k. Accordingly, if the random vector y takes values in R z, Q~,y stands for the joint distribution of x and y in R ~+z. Next, let x~, n = 1, 2 , . . . be a sequence of random vectors in RL As usual, the sequence Qx~ is called weakly convergent to Q, if