On the rate of convergence of uniform approximations for sequences of distribution functions

In this paper, we develop uniform bounds for the sequence of distribution functions of g(Vn+μn)g(Vn+μn), where gg is some smooth function, {Vn,n≥1} is a sequence of identically distributed random variables with common distribution having a bounded derivative and {μn}{μn} are constants such that μn→∞μn→∞. These bounds allow us to identify a suitable sequence of random variables which is asymptotically of the same type of g(Vn+μn)g(Vn+μn) showing that the rate of convergence for these uniform approximations depends on the ratio of the second derivative to the first derivative of gg. The corresponding generalization to the multivariate case is also analyzed. An application of our results to the STATIS-ACT method is provided in the final section.