Multiplier Central Limit Theorems

Abstract

With the notation Z i =δx i − P, the empirical central limit theorem can be written $$ \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^n {{Z_i}} \rightsquigarrow G $$ in e∞(F), where G is a (tight) Brownian bridge. Given i.i.d. real-valued random variables ξi,..., ξ n , which are independent of Z1,..., Z n , the multiplier central limit theorem asserts that $$ \frac{1}{{\sqrt n }}\sum\limits_{i = 1}^n {{\xi _i}{Z_i}} \rightsquigarrow G $$ .