A Large Deviation Principle For Moving Average Process Generated By B-Valued Stationary Random Variable Sequence

Probability Limit Theory is one of the important branches and also an essential the oritical foundations of science of Probability and Statistics. The famous probability scholars Kolmogrov and Gnedenko of 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. The large deviation principle has become one of 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.In this paper,we made a study on a large deviation principle for moving average process generated by B-valued stationary random variable sequence. We use theorem 2 in reference [1] and improve a large deviation principle of moving average process. Forα0 =1,αj = 0, j≠0, ,we can see the moving average process of theorem 3.1 becomes the random variable sequence in reference [1],so theorem 3.1 is the improvement of theorem 2 in reference [1]. Also,the establishment conditions of theorem 1.3 are satisfied in theorem 3.1 obviously, so theorem 3.1 is the improvement of theorem 1.3. There are four parts in this paper: the first part is preface considering the previous results; the second chapter includes some lemma and the proof of the lemma; the third part is the main body of the paper, consider the main conclusions, a large deviation principle for moving average process generated by B-valued stationary random variable sequence ,the last part is the prove of the theorem.First we consider the lemma as follows:Lemma 2.1 Let {Yj, 1≤i≤n} are real random variables,λi∈[0,1], 1≤i≤n, such that , thenLemma 2.2 Let {Yk,k∈Z} is a stationary random variable sequence, {bk,k∈Z} is a absolute summable sequence, n,i are any non-negative integer, thenlogEexpLemma 2.3 Let (?) is a non-negative integrable random variable which is measurable,then for anyσfield M,Lemma 2.4 Let {(?)i,i≥1} is a stationary positively random variable sequence , denote k - [n/N] ,then especially, when N=1,we haveLemma 2.5 (Proposition 0.2 on P₂₆₉ in reference[6]) (μ_ε,ε→0)satisfied the w'ULD with the rate function (?)_Y^* in (X,σ(X,Y)) field. Especially, this w*ULD is also satisfied in strong topology "s" of X, where X is a real locally convex topology vector space, Y is the subspace of topology dual space X', (?)(y) =Lemma 2.6 (04.6) on P₃₂₉ in reference[6]) Soppose convex function f : X→[-∞, +∞] have the super on a non-null open set X , then f is continuous in (domf)°,where X is a real vector space, and Domf = {x∈X|f(x) < +∞}.Lemma 2.7 Let {an, n≥1} and {bn,n≥1} are two positive real sequences,then The main theorem as follows:Theorem 3.1 Suppose {B,||·||) is a separable Banach space,{(?)k,k∈Z} is a stationary B-valued random variable sequence,satisfled isa absolute summable sequence. Define if in addition one of the following conditions holds:(i)ψ(1) <∞,and E{expθ(||(?)||)} <∞for eachθ∈R;(ii) for some n≥1,ψ+(n) <∞, and ||(?)|| is bounded,then there exist a non-negative lower semi-continuous convex function I(x),satisfied for each L≥0, {I≤L} is compact, such that (1) for each open set G (?) B,(2) for each clsed set F (?) B,that is satisfied the large deviation principle with the rate function I(x).