Two Kinds Of Convergence Rates In The Weak Law Of Large Numbers For Long-range Dependent Linear Processes

In this paper,we mainly study two kinds of convergence rates of the Marcinkiewicz-Zygmund weak law of large numbers(M-Z WLLN)in linear processes {X,n?1} with long-range dependent.The linear processes {Xn,n?1?defined by Xn =?i=-??ai+n?i for n ? 1,where {?i,-?<i<?} is a sequence of independent and identically distributed random variables with E?0=0,and{ai,-?<i<?} is a sequence of non-random real numbers.For this linear processes {Xn,n?1},for r>1,1<p<2,n r-1P(|?K=1 n Xk|>Wn(p)?)?0 as n?? for all ?>0,and for r?1,1?p<2,?n=1 ? n r-2 P(|?K=1 n Xk|>Wn(p)?<? for all ?>0,where Wn(p)=(?i=-??|?k=1nai+k|p)1/p,n?1.The two results extend the corresponding two results by Characiejus and Ra(?)kauskas[23].