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

In this paper,we made a study on a random function central limit theorem for non-stationary linear process generated by martingale differences.It in 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 limit central theory.In this paper, we improve on the strictly stationary condition of {z_j,j∈ Z} in [5].We use a similar method with [5] to obtain the same result under lim_{c→∞}sup_t∈RE[z_t²I(|z_t| > c)] = 0, and know that the stationary condition of X_t is not necessary because of theorem 2.3 and theorem 2.4.For t = 1, B — Ω.we can obtain the result of [7],and get. a similar result with [5] under the weaker condition than [7].This paper has three parts, the first chapter includes lemma and the proof of lemma 1.6 a.nd lemma 1.8, the second chapter is about a random functional central limit theorem for non-stationary linear generated by martingale differences,the third chapter includes the main conclusions in the paper.At first.we introduce the first chapter.Lemma 1.6 Let, {X_n,n ≥ 1} be a sequence of random variables on a probability space {Ω,(?), P),suppse p > 1,E|X_n|^p < ∞,n ≥ 1.ThenLet {z_t, t∈ Z} be a sequence of martingale differences defined on a probabilityspace (11 J?,P),Eiztl&t-J = 0, ï¿¡(zt2|^i_a) = a2 4 = n(ï¿¡a,)V (4)fc=l j-^-cxiLemma 1.8 Under the above conditions, let Xt = {Yl(jL-aoaj)zt< $k — E[=i Xt.Th.va. maxi]{Sk - Sk)\ = o;)(l). namely, for all e > 0.P( max is-^Sk - Sk)\ > e) -^ 0, n -> oo .l c)] = 0 . Then for B € ^"fc, A- > 1 and P(B) > 0,lim P(s;1S;,. < a:|S) = $(x) = (2tt)-5 for all x,where s^ = nol(^*L_oaa^)1. Defined for en(<) = s^^S,. + Xr+1(tn - r))nwhere r = 0,1,2, â–  â–  â–  ,n — 1.Then,all the finite dimensional distributions of â– UjiW;0 - * — l}converge weakly under the probability measure P#(-)the finite dimensional distribution of Wiener process W, namely, assume that I is all fixed positive integer ,and U,t2,-â–  â–  ,ti are real numbers that satisfy 0 < h < t2 < â–  â–  â–  < U < l.Then [)j n-oo, where PB(A) = P{A\B),At &,Theorem 2.2 Let {Xt,t G Z} be a sequence of random variables defined by (3); where {zt>^t} a martingale difference sequence and suffice the conditions in theorem 2.1. Let {yn\n 6 N} be a sequence of positive integral-valued random variables defined on the probability space (il,^.P). Also let there exist a sequence {an} of positive; integers such that an —? oo , n —> oo, and -^ -—> 9.for some real-valued random variable 9 with P(0 < 9 < oo) = l.Theiuthe process {^n(i),0 < / < l}converge weakly to the Wiener process W, whereU(t) = s;^(ST + XT^{t.un - r)) , - c)\&t-i] = 0 , a.s, c^*rx tezThen, for all fixed A; > 1,B e &k. and P{B) > 0,liin is^Sn < x\B) = *(x) = (2tt)⁵ f J —O...