Limit theorems of general functions of independent and identically distributed random variables

In this paper we derive limit theorems of some general functions of independent and identically distributed random variables. A stability property is used to derive the limit theory for general functions. A procedure followed in de Haan (1976) and Leadbetter et al. (1983) is used to prove the main result. The limit theorems for the maximum, minimum and sum of fixed sample sizes and random sample sizes are derived as special cases of the main result.