Abstract:

Let {ξ1,ξ2,…,ξn} be a sequence of indepen
dent and identically distributed U［0,1］distributed random variables. The
 uniform empirical process generated by them is defined as Fn(t)=n-1/2∑
〖DD(〗n〖〗i=1〖DD)〗(I{ξi≤t}-t),〓 0≤t≤1; 〓‖Fn‖=sup〖DD(〗
〖〗0≤t≤1〖DD)〗〖JB(|〗Fn(t)〖JB)|〗.Let U be the Brownian bridge of
 D［0,1］ and ‖U‖=sup〖DD(〗〖〗0≤t≤1〖DD)〗〖JB(|〗U(t)〖JB)|
〗. With the strong convergence of probability, we obtained the f
ollowing almost sure central limit theorem for ‖Fn‖ and sup〖D
D(〗〖〗0≤t≤1〖DD)〗Fn(t):
lim〖DD(〗〖〗n→∞〖DD)〗〖SX(〗1〖〗log n〖SX)〗∑〖DD(〗n〖〗
k=1〖DD)〗〖SX(〗1〖〗k〖SX)〗I{‖Fk‖≤x}=P{‖U‖≤x}=1+2∑〖DD(〗∞〖〗k=1〖
DD)〗(-1)ke-2k2x2〓a.s.[HJ]

Key words: uniform empirical process, almost sure central limit theorem, Brownian bridge