The functional law of the iterated logarithm for the empirical process based on sample means

Consider a double array {X-i,X- j ; i greater than or equal to 1. j greater than or equal to 1} of i.i.d. random variables with mean mu and variance sigma (2) (0 < sigma (2) < infinity) and set Z(i, n)= n(-1/2) Sigma (i)(j=1) =1 (X-i,X- j - mu), sigma. Let <(<Phi>)over cap>(N, n) we denote the empirical distribution function of Z(1,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 rootN(<(<Phi>)over cap>(N,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 al e also presented.