Moderate Deviation Principles For Moving Average Processes Of D-Dimensional Stationary Sequences

Probability limit theory is one of the important branches and also an essential the oritical foundations of science of probability and statistics.Moderate deviation principles is a pop topic of the probability research in recent years. One of the difference between it and other limit theories (such as strong law of large numbers,central limit theorems,complete convergence,invariance principle,the law of the iterated logarithm and so on) is that it has rate function I.And it's so interesting that in different moderate deviation the rate function is kinetic energy or interaction energy.We can say that I give moderate deviation more abundant intension compared usual limit theories.So people pay more attention to it.Though there are almost a little report about the quality of moderate deviation for the moving average processes of d-dimensional random variables sequence with stationary distribution. Some research and results had been given in this dissertation.First,let's give the definition of moderate deviation principles for the summation of d-dimensional random variables.Suppose {X_k,k≥1} is a d-dimensional random variables sequence,let,n≥1.Suppose {B_n, n≥1} is a series of positive real numbers,satisfiesI(x) is a nonnegtive lower compact function (namely (?)L≥0, {I≤L} is a compact set) .It's said that {P ( S_n/B_n∈·) , n→∞} satisfies the moderate de-viation principles (MDP in short) with the rate function I(x).If for any open set G∈R^d ,closed set F∈R^d,This paper has four parts,the first part is preface considering the previous definition.The second chapter includes lemma and the proof.The third part is the main body of the paper,consider the main conclusions, the moderate deviation principles for the moving average processes of d-dimensional random variables with stationary distribution.The last part is the prove of the theorem.First we consider the lemma as follows:Lemma 2.1 Suppose {Y_i, 1≤i≤n} is a real random variables sequence,λ_i∈[0,1], 1≤i≤n, such thatsoLemma 2.2 (Literature[8]P₂₇₁Proposition 1.1) Suppose {μ_ε,ε> 0} isprobability measure in (R_d, (?)(R^d)).Let∧(y) (?) lim supεlog,(?)_y∈R^d,∧^*(x) = sup((x,y) -∧(y)|y∈R^d),(?)x∈R^d.Then the following properties are equivalence:(a) {μ_ε,ε→0} satisfies the ULD with the rating function is given by∧^* in R^d;(b) (?)x∈R^d, (?)δ> 0,such that∧(δx) < +∞; (c) (?)δ> 0,such thathere ||x|| = |x₁| + |x₂|+…+|x_d|.Lemma 2.3 Suppose {X_k,k≥1} is a d-dimensional random variables sequence.{B_n,n≥1} is a series of positive real numbers,satisfies The definition of function∧(y),∧^*(x) have been given. Ifsatisfies the moderate deviation principles withthe rate function∧^*(x).Then there is a r₀ > 0,such that for any y∈U(0, r₀), (here y∈R^d), we all have∧(y) <∞.Lemma 2.4 Suppose {Y_k, k∈Z} is a d-dimensional random variables sequence with stationary distribution.{b_k,k∈Z} is a, absolute summable sequence of real numbers.ThenLemma 2.5 Suppose {a_n,n≥1} and {b_nin≥1} are two positive real sequences,thenThe main theorem as follows:Theorem 3.1 Suppose {ξ_k,,k∈Z} is a d-dimensional random variablessequence with stationary distribution.δ₀ > 0, such that Ee^{δ₀||ξ₁||} <∞, {B_n, n≥1} is a absolute summable sequence of real numbers ,satisfies satisfies the moderate deviationprinciple with the rate function∧^*(x).Let {a_i, i∈Z} is an absolutely summable sequence of real numbers.Define k≥1.Then there is a nonnegative lower-compact function I(x), such that {P (S_n/B_n∈·) ,n→∞} satisfies the moderate deviation principle with the rate functionI(x).Here...