A uniform law of large numbers and empirical central limit theorem for limits of finite populations

Abstract

Let U k be a finite poulation of size N k with elements a kj , 1≤j≤N k , all contained in a space U, and let (Y k1 , ..., Y k,Nk ) be one of the N k ! permutations of (a k1 , ... a k,Nk ) chosen with equal probabilities. The vector (Y k1 , ..., Y k,nk ), n k ≤N k , is an (ordered) simple random sample (SRS) from the population U k . Our main results are a uniform law of large numbers and an empirical central limit theorem for the array {Y kj : 1≤j≤n k }