Let {Xi:-∞I<i<∞} be a Φ-mixing stationary sequence of random vectors. Let Fn(x) be the corresponding empirical distribution function of X1,…,Xn,and let X ,be the sample mean of the present investigation is to show that the asymptotic almostsure representation of Fn(X). Two different orders of the remainder term,under different Φ-mixing conditions, are obtained and used for proving functional central limit theorrms, laws of iterated logarithm, Wiener processes embedding and invariant principles for Fn(X).
