Nonlinear Extensions of a Limit Theorem,

A sequence of random variables xn is given together with centering and normalizing constants an and bn, and it is assumed that the distribution of xn-anbn coverges weakly as n approaches infinity to distribution F. The purpose of the present paper is to extend this convergence property to the sequence Vxn where V is a given linear function. Specifically it is desired to find centering and normalizing constants alpha n and beta n such that the distribution of Vxn-alpha nbeta n converges weakly to a limit G and to evaluate this limit. Sufficient conditions are presented that ensure the existence of the limit G, and it is shown that G is simply related to F. To illustrate the variety of limit laws that can arise, several examples are considered. Author