A Random Functional Central Limit Theorem For Linear Process Generated By Martingale Differences

In this paper we made a study on a random function central limit theorems for linear process generated by martingale differences. It is well known that the central limit theorem is one of the most important results in the probability theory, and is applied in a lot of scientific studies. Furthermore, with the development of economy and the advancement of society, there is the more deep study about the central limit theorem. In this paper, we improve on the strictly stationary condition of {z_j, j ∈ Z}. We use a similar method to obtain the same results under E[z_t~2I(|z|_t > c)] = 0, and known that the sta-tionary condition of Xt is not necessary because of theorem theorem 3.3 and theorem 3.4.This paper has three parts, the first chapter is about a result on asymptotic normality for series sum. The second chapter is about a, random function central limit theorems for linear process generated by martingale differences. The third chapter is about a random function central limit theorems for non-stationary linear process generated by martingale differences.The chief results of the first chapter are as follows:Theorem 1.1 Let {X_t} be a sequence of stationary random variables defined by (1.1.1), where {Z_t,F_t} is a martingale difference secquence, assume that (1.1.2) and (1.1.4) hold,andThen, we have (1.1.3).The chief results of the second chapter are as follows:Theorem 2.1 Assume that (2.1.1) and (2.1.2) hold. Then ,for all fixed k and B e J"fc,P(B) > 0,lim PirT^'in—?-oon 1} be a sequence of random variablesdefined (3.2.3). Sn = ￡ Xt, where {z^T^t G Z} is a martingales differencet=i sequence, assume that (3.1.2) and (3.1.6), andUrn sup E[z?I(\zt\ > c)] = 0. Then for B €Tk,k>\ and P(B) > 0,tfSn < x\B) = ?(x) = (27T)-1/OOfor all x, where s2n = ncr2( ￡ a^)2. Defined forJ=-00tn-r)), L?■**? *i are real numbers that satisfy 0 < ti < t2 < ??? 1} be a sequence of random variables defined (3.2.3). where {zt,Tt,t € Z} is a martingales difference sequence and suffice the conditions in theorem 3.1. Let {vn;n € iV} be a sequence of positive inter-valued random variables defined on the robability space (f2, T, P). Also let there exits a sequence {an,n > 1} of positive integers such that an -> oo, n —> oo and ^ A^, for some real-valued random variable 0 with P(0 < 9 < oo) = 1. Then, the process {^vn(w),0 < u < 1} converge weakly to the Wiener process W,whereU(t) = s-n\Sr + Xr+1(tvn - r)), - < t < T—-,vn vnloTSn=￡xi,sl = no2{ g aj)2,r = 0,l,2,..-,t;n-l.t=l j=—ooTheorem 3.3 Let {Xt,t > 1} be a sequence of random variablesndefined (3.2.3). Sn = X) Xt, where {zt,JTt,t € Z} is a martingales difference sequence, assume that (3.1.2) and (3.1.6), andlim swpE[zp(\zt\ > c)|^i-i] = 0,o.s. c~+0° tzzThen for all fixed k > 1,B G J^, and P(B) > 0,Jim00for all x, where s2 = na2( ￡ Oj)2. Defined forj=—ooUt) = ^X(5r + Xr+l(tn - r)), ^ < t oo, where PS(>1) = P(A\B),A € ^.Theorem 3.4 Let {Xt, t > 1} be a sequence of random variablesdefined (3.2.3). Sn = ￡ -^t, where {^t,^*, t € Z\ is a martingales differencet=i sequence and suffice the conditions in theorem 3.3. Let {vn;n € N} be asequence of positive inter-valued random variables defined on the robability space (Q, F, P). Also let there exits a sequence {an, n > 1} of positive integers such that an —> oo, n —> oo and ^ A 9, for some real-valued random variable 0 with F(0 < 6 < oo) = 1. Then, the process {￡??(?), 0 < u < 1} converge weakly to the Wiener process W,wherefor 5n = EXi,s2=ncr2( E o^.r = 0,1,2,-??,??-1.t=l j=-oo...