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+nζi for n ≥ 1,where {ζi,-∞<i<∞} is a sequence of independent and identically distributed random variables with Eζ0=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)ε)→0 as n→∞ for all ε>0,and for r≥1,1≤p<2,∑n=1 ∞ n r-2 P(|∑K=1 n Xk|>Wn(p)ε<∞ for all ε>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].