Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences
Jiang Li · 2005
Let {X,X_n;n 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances.Set S_n=∑_(k=1)~n X_k,M_n=max_(k n)|S_k|,n 1.Suppose lim_(n→∞) ES_n~2/n=:σ~20 and ∑_(n-1)~∞ ρ~(2/d)(2~n)∞,where d=2 if 1 r2 and dr it r 2.We prove that if E|X| ∞,for 1 p2 and rp,then e~(2(r-p))/((2-p))(sum from n=1 to ∞)n~(r/p-2)P{M_n en~(1/p)}=(2p)/(r-p)(sum from k=0 to ∞)((-1)~k)/((2k+1)~(2(r-p/(2-p))))E|Z|~((2 )/(2-p)) where Z has a normal distribution with mean 0 and variance σ~2.