Precise Asymptotics in the BaumKatz and Davis Laws of Large Numbers of p-mixing Sequences

Let {X,X n ; n . 1} be a strictly stationary sequence of p-mixing random variables with mean zeros and finite variances. Set Sn=.nk=1Xk,Mn=maxk.n|Sk|,n.1. Suppose limn..ES2n/n=:.2>0and ..n=1.2/d(2n)<.,where d = 2 if 1 . r < 2 and d > r if r . 2. We prove that if E.X.r < ., for 1 . p < 2 and r > p, then lim....2(r.p)2.p.n=1.nr/p.2P{Mn..n1/p}=2pr.p.k=0.(.1)k(2k+1)2(r. p)/(2.p)E|Z|2(r.p)2.p, where Z has a normal distribution with mean 0 and variance .2.