The uniform convergence of the sum of independent random variables

LetX_i (i=1, 2, ...) be the independent random variables on the probability space (, ,P). In this paper we will show that the necessary and sufficient condition of the uniform convergence of X_i is the convergence of m_i⁽¹⁾ and m_i⁽⁰⁾, wherem_i⁽¹⁾, m_i⁽⁰⁾ denote the 1-quantile and 0-quantile ofX_i respectively.