均匀经验过程几乎处处中心极限定理的一个注记

A Note on Almost Sure Central Limit Theorem for Uniform Empirical Processes

设{ξ1,ξ2,…,ξn}为来自[0,1]上服从均匀分布的独立同分布样本,产生的经验过程为Fn(t)=n-1/2∑ni=1(I{ξi≤t}-t),0≤t≤1;‖.‖表示一致模,即‖Fn‖=sup0≤t≤1 Fn(t);U为D[0,1]上的Brown桥,‖U‖=sup0≤t≤1 U(t).利用概率强收敛工具,得到了关于‖Fn‖及sup0≤t≤1Fn(t)的形如nl→im∞lo1g n∑nk=11k I{‖Fk‖≤x}=P{‖U‖≤x}=1+2∑∞k=1(-1)ke-2k2x2a.s.的几乎处处中心极限定理.

Let {ξ1,ξ2,…,ξn} be a sequence of independent and identically distributed U-distributed random variables.The uniform empirical process generated by them is defined as Fn（t）=n-1/2∑n i=1（I{ξi≤t}-t）,0≤t≤1;‖Fn‖=sup 0≤t≤1Fn（t）.Let U be the Brownian bridge of D and ‖U‖=sup 0≤t≤1U（t）.With the strong convergence of probability,we obtained the following almost sure central limit theorem for ‖Fn‖ and sup 0≤t≤1Fn（t）：lim n→∞1 log n∑n k=11 kI{‖Fk‖≤x}=P{‖U‖≤x}=1＋2∑∞ k=1（-1）ke-2k2x2 a.s.