The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array {Xi j; i>1, j>1} of i.i.d. random variables with mean (mu)and variance (sigma)^2 (0<(sigma)^2<(infinity))and set z= n-1/2 (sigma)=1(Xi,j – (mu))/ (sigma). Let (phi) denote the empirical distribution function of Z1, n ,..., Z N, n and let (phi) be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for (sigma) N ((phi)(N)- (phi)), where n=n(N)(infinity) as N-(infinity). For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.