An estimate concerning the Kolmogoroff limit distribution
Kai-Lai Chung · Transactions of the American Mathematical Society · 1949
We consider a sequence of independent random variables having the common distribution function F(x) which is assumed to be continuous.Let nFn(x) denote the number of random variables among the first n of the sequence whose values do not exceed x.Write (1.1) dn= sup (Fn(x) -F(x))\.-W <iC<e© Kolmogoroff [l](2) proved that the probabilitywhere X is a positive constant, tends as »-»co uniformly in X to the limiting distribution (1.3) *(X) = X(-l)'6-2'v.-oc