A Large Deviation Principle For Moving Average Process Generated By φ-Mixing Stationary Random Sequence

In this paper,we made a study on a large deviation principle for moving average process generated by finite dimensional φ—mixing stationary random variable sequence.We use theorem 1 in reference [1] and improve a large deviation principle of moving average process.For a₀ = l,a_j = 0,j ≠ 0,we can see the conclusion of theorem 3.1 is the same of lemma 2.1,so theorem 3.1 is the improve of theorem 1 in reference [1].This paper have three parts:the first part is preface considering the previous results;the second part is the lemmas includes lemmas and the proof of we needs;the third part is the main body of the paper.First we consider the lemmas as follows:Lemma 2.1 Let {ξ_κ,κ ∈Z} is φ- mixing stationary random sequence in R^d, ||ξ₁|| ≤ C < ∞, satisfies for each K≥ 0, e^knφ{n) → 0, (n→∞). let Then there is a non-negative convex lowercompact function I : R^d →[0, ∞], such that(1) for each closed set F c R^d,(2) for each open set G C R^d,(3) for Vy G Rd, limitA(y) = lim - log E exp n—?oo fiexist, and the rate function is given byI(x) = sup{< x,y > -A(y) : y G Rd},here < x,y > is the interior product of x,y.Lemma2.2 Let Domf = {x e X\f(x) < +00}, (Domf)° is a set of interior points, if convex function / : X —> [—00, +00] have the super on a non-null open set X, then / is continuous in (Domf)°.Lemma2.3 Let {Yi, 1 < i < n} is real random variables,Aj Gn[0,1], 1 < i < n, such that J2 K = 1) thenlog E exp I Y2 KYi \ < 53 Ailog E exp Yt. U=i J t=iLemma2.4 Let {yuï¿¡,e > 0} is probability measure in (Rd, B(Rd)), let A(y) = lim sup e log/fjd exp(< x, y > /e)dfxï¿¡(x), Vy G i?d,A*(a;) = sup(< x,y > —A(y)\y G Rd),Vx G i?d. Then the following properties are equivalence:(a) {//ï¿¡,ï¿¡ —> 0} satisfies the ULD with the rating function is given by A* in Rd;(b) \fx G Rd, 36 > 0, such that A(5x) < +00;(c) 36 > 0, such thatlimsupsrlog / exp(5\\x\\/e)dfiï¿¡(x) < +00.here ||x|| = |xx| + \x2\ +-----1- \xd\.Lemma2.5 Let{an, n > 1}, {bn, n > 1} are two real sequences,thenlim sup - log(an + bn) < lim sup - log an \f lim sup - log bn,n—>oo 71 n—>oo 71 n—>oo 71lim inf — log(an + bn) < liminf — log an \J limsup — log bn.n^oo n n—oo n v ^oq nThe main theorem as follows:Theorem 3.1 Let {&, k G 2f} be a (^-mixing stationary random variable sequence in Rd, ||&|| < C < oo, i G Z,for each K > 0, eKn(j){n) —> 0, (n —> oo). Let {a^i G Z} be a absolute summable sequence of realoonumbers, let Xk = X) ajï¿¡k-j> k > 1. Sn = X) -Xfc>^ > 1- Then there isj=-oo fc=la non-negative lower compact function I : Rd —> [0, oo],such that (1) for each closed set F C Rd,l P flim sup-log P f— ef)<- inf I(x),n->oo 72 V n / xeF(2) for each open set G Clim inf - log P (— G g) > - infn-?oo 77, \ 77, / ï¿¡GThat is {P f^ G ?) , n -^ oo} satisfies the large deviation principle with the rate function is given by I(x).