The Precise Asymptotics Of The Complete Convergence For M-Dependent Banach Space Valued Random Variables

Probability Limit Theory is one of the important branches and also an essential the oritical foundations of science of Probability and Statistics. A famous probability scholar from previous Soviet Union said :"Only Probability Limit Theory can reveal the epistemological value of Probability . Without it,you couldn't understand the real meaning of the fundamental conceptions in Probability." Classical limit theory is the signify achievement in the process of Probability. Strong convergence has become the most important and popular orientations of the current study of Probability Limit Theory. Some significant results have been reached through deep research in this dissertation.Let {X, X_n,n≥ 1} be a sequence of i.i.d.random variables. Hsu and Robbins (1947) first brought forward the definition of complete convergence:holds, if and only if EX = 0, E|X|² < ∞.Erdos (1949,1950)andSpiter (1956) discussed corresponding results. In the 60's of the 20th century, Katz(1963)Baum and Katz (1965)extended their results: Let 1 ≤p ? 2, r≥ p, thenholds, if and only if E|X₁|^γ < ∞, and, when r≥ l,EX₁ = 0.Davis (1968) proved:for any e > 0,P{\Sn\ > ey/nlogn} < oo.holds, if and only if EXi = OandE'Xf < oo.Many authors considered various of the results of Hsu-Robbins and Baum-Katz.Sp&taru(1999) studied the precise asymptotics of the infinte sums as e -$ 0: Let {X,Xn,n > 1} be a sequence of i.i.d.random variables with 2 < oo,Sn = ï¿¡) Xjt then,I OO 1lim —-----T, -P{\Sn\ > en} = 2.This paper has four parts, the first part is preface considering the previous results. The second chapter includes lemma and the proof of Iemma2.4. The third part is the main body of the paper, consider the main conclusions, the precise asymptotics of the complete convergence for m-dependent banach space valued random variables.The last part is the prove of the theorems.First we consider the lemma as follows:Lemma 2.1 Let Xis the B valued variables, and X is the symmetry of X, then for any t, a > 0,P{\\ X ||< a}P{\\ X ||> t + a} < P{\\ X ||> t}. assuming that P{\\ X ||< o} > l/2,then,and for 0 < r < oo, E \\ X \\r< oo,if and only if E \\ X ||r< oo.Lemma 2.2 Let{X,,i > 1} be a sequence of i.i.d.random variables with B valued. Then for the symmetry sets{Ai, 1 < i < n},wo haveLemma 2.3 Let Z\, Z2) ? ? ?, Zn be a sequence of i.i.d.random variables with B valued. And for some q > 2,E{\\ Z{ \\q) < oo, 1 < t < n. Then for t>0,i 11} ^ expi-tyilUA^+C^EiW Z, \\')lt\j=X j=l j=l(2.7) Where An := sup{Y, E{{Zj,y)2} :|| V ||< 1}, Cidepending only on q.Lemma 2.4 Assumed that {Sn/y/ri,n > 1} convergence weakly to G with B valued. LetAn = sup | P{\\ Sn || /y/n >x}- P{\\ G j|> x} |,then we havelim An = 0.n-+ooLemma 2.5 Let {Xn,n > 1} be a sequence of m-dependent random variables with B valued, Assumed that X\ e CL(B),then we have (l)For any / e B*, we have Ef{Xx) = 0, Ef2{Xx) < oo;(2)limA2P{||X1||>A}=0;(3)sutp sup C2Pin'1/2 || E Xi || > C} < oo;n€N OO t=l(4)For any 0 < P < 2,we have(5)Suppose that for any / e B',f ^ 0, we have f e B*,f ^ 0, then there exits the central measure (2 with B valued such that {Sn/y/n, n > 1} convergence weakly to G,andm+lCovaifJ) = miEfiXJ + 2 ï¿¡ Ef{X1)f{Xk)).k=2The main theorem as follows:Theorem 3.1 Supposed that E || Xi ||r< oo, r > l.then for 1 < p < 2, r > p,we have) f]nr/P-*p{\\ Sn \\> en1'"} = -^—E II G ll^*^f J r — pTheorem 3.2 For 1 < p < 2, EXX = 0, E || Xj ||2= a2 < oo, lim -=i— ï¿¡ -P{\\ Sn \\> e ^2-p Theorem 3.3 supposed that JB^ = 0, E \\ Xx ||2< oo,for 0 < 5 < 1,n=l...