Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process

Let {ξ1,ξ2,…,ξn} be a sequence of independent and identically distributed U[0,1]-distributed random variables. Define the uniform empirical process Fn(t)=n−12∑i=1n(I{ξi≤t}−t),0≤t≤1,‖Fn‖=sup0≤t≤1|Fn(t)|. For 1≤p<2,an=O(1/loglogn), we have the limits of weighted infinite series of P{‖Fn‖≥εn1/p−1/2}, P{‖Fn‖≥εlogn} and P{‖Fn‖≥(ε+an)2loglogn} as ε→0.