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(1)'. The Rate of Convergence to Normality for Strong Mixing Sequences of Random Variables By. F. Hiroshi NEGISHI* '. gl. Introduction. Let {xb -oo<i<oo} be a strictly stationary process with E(xj)==O, satisfying the strong mixing condition, i, e.. '. Sup IP(AB)-P(A)P(B)I=cr(n)->O (n-oo). AEM"... BGMaOe+n. where Mg denotes the o-algebra generated by {xj,1'=a,･･･,b}. Write S.==xi+ ･･･ +x. and a2==E(x,)2+2 ]E)ee..i E(x,xj).. The following theorems are presented by Ibragimov (cf. [2]); 22:E.M,,A,',,,I,lleiEf[.X'l .2<T .H. 2o',5ZhZ?. fbr SOMe 5>O a"d if ;.)(a(1'))6ie'6'<oo, then. P(S./a4vl-ff < z) . ￠(2) as n-> oo, where di(z)== v1 T SZ-.e-t2i2dt. 2-. THEoREM B. Il/ x7s are bounded, i.e. Ixjl<C<oo with Probability 1 and if Za(n)<oo, then a2<oo and moreover if o>O, then. P(S./oV"ff <z) . di(z) as n- cx). ' ' The process {xj} is said that it satisfies the absolute regularity condition if. P(n)=E(SuplP(AlMO-..)-P(A)l)SO (n-,-oo). '. Aeilf;7 ' '. Since a(n)$P(n), it follows that if {xj} is absolutely regular, then it is strong. mlxmg.. Recently, in [5], Yoshihara proved the following theorem, by his useful method. " Department of Mathematics, Faculty of Education, Yokohama National University..

(2) 18 ･ H.NEGIsHI. '. THEoREM C. Let {xj} be a strictly stationary, absolutely regular Process. with E(xj)=O and EIxjl`'6<oo for some S>O. if P(n)=O(e-rn) for some r>O and o>O, then SupIP(S./o V-T < z) -￠(2)I= O(n-ii`(log n)3i2).. z. '. ,. In this paper, we shall estimate the convergence rate to normality for. somestrongmixingprocesses,usingthefamiliarway. , t. g2. Theorems.. Jt ' ' In what follows, we assume that a>O. Our results for strong mixing processes are the followings.. ･ THEoREM 1. Let {xj} be a strictly stationary, strong mixing seqnence of random variables with E(xj)=O. ILf EIxjl`<oo and ￡(cr(n))ii2(2"6)<oo for some. 6;;}iO then ' ' n. -7. SUP]P(S./a V-iT < 2) - di(2)1=O(n-(i"6)14(i5+86).. z. THEoREM 2. Let {xj} be a strictly stationary, strong mixing seqnence of random variables with E(xj)==O. If x/s are bozanded, i.e. Ixjl<C<oo with Probability 1 and if 2(cr(n))iiì'6'<c>o for some S>O. then. n O(nHO"i`logn) (O<S:1/2),. sup[P(Sn/aV-iT<2)Adi(2)1='J{o(.-ys) (1/2<S)･ '. g3. Proof of Theorems. /tt (cf. t [4]). In the ' prove both theorems by the well known method We shall following, we shall denote by the letter K, with or without indices, positive CQnstants.. ' ' PROoF oF THEoREM 1. At first, we remark that E(S.)2=a.2=na2(1+o(1)).. Let .. fiv(x)={g ,(ii,itilil￡I)'. andf-N(x)=x-fN(x). Define Sn(N) = ..vt'ii' ,=, ln (.1`liv(Xj)-E(f}v(x,)). l.

(3) ' The Rate of Convergence to Normality 19 ' ' Sn(N)=.Jl>-iT,.-n,(f-N(xj)-E(f-N(xj)). ' For a small e>O (in fact e<1/14), put. N=ne, p==p(n) =[nli2;e], q==q(n)==[nii2-e], le=le(n)=[plq]. and put ' '. 1 k-ip T"== a.vl-ii-,,Z=, ,2.=,(-7Civ(Xt(p+q)+J)-E(.ICzv(Xz(p+q)+J))) - / '. ' '. Th == Sn(N)mT"= lk ...i-ii- i･Il=oCi ' where. q Ci=･ 2 (.1ÌAr(Xi(p+q)+p+J')-E(.11ivr(Xi(p+q)+p+j')) j'=1. ., . (i=O, 1{ ･I･, le-1) . Ck=T-j.=fe(pZ"+q)+i{.IC12v(xP-E(.Lv</2))}･ ' ,,.,.,.,.,. '. Then,fromthestrongmixingcondition,wehave･' ' ''. (1)E(g.(N))2=l{E[f-.'(.,)LE(f-.(.,)>]2 .' '.a '. ' ,. +2¥,i(17--;l;)E[f-N(x,)LE(f-'N(xo))][f-N(x,)-E('f-N(xj))]} ;;l ;, {Elf-NSxo)]2+16[I!lIf-N(x,)l`(2'O")!(3'26)](3+2o")i2(2+o"). ℃¥i, (1-i' )(a(1))ii2(2+s)}. ;:iiil,7{[7ill;T,Elf-N(x.t')[` .., , ..- ,.', . ' ...:,.)6,,lll.,',i(l!'5':10,if.E,li.:.5,gr'i`)(3'2o'i2(2"6) :;s.t (i-f/? f.,,,,.,,,:,,.}. Since a(j)=o(1-2(2'b)), i.rom the, proof of Lem.ma 18 5 2 in [3] .. q E[Cel`$KiN`q2,2...,jcr(1')SK,N`q2max(1,q-2(i'o-)). .,..,..... r '. '. '.

(4) 20 . H. NEGIsHI and so. (2) E(Ta)2= 1,E(le2'i. no i=o. c,+c,)2. 1 k-1 k-1. ;!ll na2 {leE(Co)2+2le i=, IE(CoCi)l+E(Ck)2+2lE(kCh) ,l=,. 1}. 1 le. ;III ..2 {kE(Co)2+16le(ElCol2)ii2(El4ol`)ii` ,l.X=,(cr(iP))ii`+E(4le)2. le-2 . +16(E1C,l4)ii4(EI4,l2)ii2Z(cr((le-i)p))ii4+2(EZ,-,]2･ElC,12)ii2}. i=o. $ -ril.}2 {lea3+ka,NV-4 e.,(a(i))'i`+a2.+,+NV'a-a,.,,l{.l, (a(i))ii`+ o,a,.,}. $ .52 {leq+leqAi)-6!2+(p+q)+NVq(p+q) p-6i2+ Vp(p+q)} ;SI<bn-(2eA(E(1-r612)+614))-<.o(n-e(1+al(2+6))).. '. Since. lE(eit8nlaVft)-E(eitT.)[ .<.,lE(ei"Sn'aVn-)-E(ei"Sn(N))l+IE(eitSn(N))-E(eZt'n)l :;l ltl{'ViE1g.(Ai')l2 + ･vistI TTI'ITi2-},. we obtain from (1) and (2). '. (3) I,=snLl E(eitSn!aVfil-E(eicTn) Idt. -n. r. ;:S S" , { 'ViEIg.(N)l2 + iviEiTa[2} dt==o(n-(ei4)(i+6i(2+6))). , -.n. where r=t(1+2$) Since an2=:n{o2-2j...E(xoxj)-l]IIIi1'E(XoXJ)} we have (4) -li[/S22m1i:ilrl;,T{j-..IE(xoxj)l+-ill-7i]'[E(xoxi)l}. $ -jl,T 'ViE1xol`{ j-.. (cr(j'))i/2+ -ill' ]l.n.,' ]'(cr(]))i!2} SKn-i. and so from (1), (2) and (4).

(5) The Rate of Convergence to Normality. 21. (5) IE(Tn)2-ll;ll E(Tn)2- .a.Z2 + .a32 -1. '. 521E(T.(T"+SP.(N)))l+E(T"+g.(N))2+Kn-i ;ll2･vi ElT. I2(･v!ElT"l2 + 'V'.E lg.(N)l2). +2(ElT"I2+ElS.(N)l2)+Kn-i 40(n-(e12)(i+tif(2+o"))). Now, let 6o, &, ''', ek-i be independent random variables distributed in the. same way as the corresponding. lp. a .v"ii' j=i (j`li"(Xt(p+q)+j) -E(.7Qv(x,(,.,).,)). (i=O, 1, ･･･, le-!). From the strong mixing condition. k-1 ;S4ka(q)=leO(q-2(2'fi)) ll E(e"i') E (eitT.)m J'=O. ='O(n"(112-E)(3+26)). On the other hand E(eitTn)- ,len.li E(ei`"'j-) ;si -S2L(ElT.I2+feE(6,)2). for all sufiiciently small t. So we have. k-1. (6) I,..Snr , E(ei`Tn)- tE(e`tj`Z-o6 b dt. -n. le-1 k-1. ;:llS],i$..ii, E(eitT")-tE(eitj'￡=OS') dt+S.-y,s,]s.r E(eitTn)-Ill(eitj'Z=o6j'). '. ;-:!{"ll-(E1Tn12+feE(go)2)S,is..,1,ltldt+O(n-(i12-S)(3'20"))S../4si,is.riEltfi'tl. = O(n-(ef4)(i+61(2+O))).. Furthermore, let , C'==.v'E{'i.)2 (1'=O,1,･･:j'leLbl'''i. Then, by the analogous argument to (3), we have from (5). le-1 le-1, .r E(eitj'Z=oS')-E(eitJ'￡=oS'). (7) -nI3=S, .t dt. dt.

(6) 22 H. NEGisHi ;;l2nrEle-1 Z (6j-C･) $2nr1VE(T.)2-11. j'=o :. ;;l2nr1E(T.)2-11==O(n-(E14)(i+o"1(2+6))).. Finally, by applying Esseen's lemma to the sum l2. ]ll C･, we have. le-1 , 1E(eZ`2io6')-2-t2121;:llKble-il2p,,lelt13e-`2i` le - 1/2. if 1t1;Sl 24p,,,, where p3,k==EIe,13/(E(6,)2)3i2.. Since E(8o)2 =a2P(1+o(1)) and. ' .Ele,f4,<..KN`P2maox,il,p-2(i+6)). we have. ' ･. p3,le;$[EE2fo,l,`))l,iisK,N3=-K,n3e,.'. and consequently,. k-1 , (s) IE(eitJi-oS)le-t2121$Kn-(1/4"(712)e)lt13e-"t214 ., ,. .,,, holds for all suMciently large n and for all t.such that: ltlSKni!`-(7i2)e. Taking eo satisfying {t' (1+ 24s)==t--il-eo for some Slo, we have from (s). k-1 , (g)I4..snrOroE(eitj'Z=06;)-e-t2i2dt;sKn-(ii4-(7i2)Eo), '':''..･/.'･. wherero==. -n ' ･' t'(i+.226)'' i. Combining(3),(6),(7)and-(9),fromEsseen'stheorem ,' (10) SuplP(S./o'･vi-ii-<2)-CZ)(2)1.. i''' SK,SZr.Or,E(ei"8"ia.Vt"-)'e-t2i2dt+K,n-'r,. . SKo(Ii+I2+I3+I,)+K,n-ro==O(n-ro). '. This completes a proof, of the theorem.. pRooF oF THEoREM 2. The proof'is carried 6nf by the same way of the. proof of Theorem l. ' ･.･ ' l･･ p==[n3i`],q=[nii4],'le=[plq'] ' .'. '1,, '.

(7) The Rate of Convergence to Normality. 23. and put .,. Tn=:.,i&, T"=.vSt"-iT-Tn==.im. tk.oC` where. lp '. 6i=a.vi-iT}･lll=,Xi(p+q)+j (i=O,1,'",le71). e ' p ."i==Zxi(p+q).p.j (i=O,1,･･･,le-1) J=1 n. C le = j･= le(pZtq)+1 Xj''. Since a(i)==o(i-("6)) and a.2=no2(1+o(1)), we have. ' ' tk.,ev(iP)+a}+q (11) ,E(Ti)2;;l .1.2 {ka2p+16C2q2le +16C2q(p+q) :.i, a((le-i)P)+2a,a,+,}. gKn-r .. -. where r=min(-ll-, 1+435,26). , From (11), we have. ' (12)snr/ f,E(ei`Snia .V"' t)-E(eitTn)dt:llKIn-ri4.. . r`. -n. WehaVe. //.,'Ll$2.c,2{,-.i.(7-)+2i-ZEi,j'cr(i)}$Kbn-6. and so from (11) .2. .2. (13) IE(Tn)2mll;Sl E(Tn)2- .a ", + .a ", -1. iS. i. 5Kn-r/2+K6n"6.. '. Since .,,,... ' E(ei`T")-jl]il,E(ei`S')ls4kcr(q)fo(n-o"14).. tt. tt it is easily obtained by the same way as in (6) that. le-1 r14 E(e"'n)-IflIE(eiteJ). (14)Snri, te'==O dtSKn'o"i41ogn. -n. .l,.

(8) 24 H. NEGIsHI From (13) we obtain. r/4 k]']IIE(eitg"j-).leI'IIE(eitg"J'IVE(T.)2 ' '. (15)S"rk"=O "=Ot. dt$K,n-ri4+K,n-(6-ri4). -n Finally, let 66, 61,･･･,6-i be independent random variables distributed in. the same way as the corresponding &/VE(T.)2. From Esseen's lemma. h-1 , E(eZtjZ..ogJ)-e-t2!2 $Kole-112p3,le1t13e--t214 if 1tj S. tV le/24p3,le, where. E1g, 13. P3,le = (E(6,)2)312 SKi･. Therefore, ' (i6) snr'. le-1 ,. i, E(eitj'Z=Oe. r`. tj'. )-e't2i2 dtsKle-,i2.,,Kn-iis,. -n if 2-s{; -IL.. 4-8 Combining (12), (14), (15) and (16), from Esseen's theorem. ' SuplP(S./a･v"-iT<2)-di(z)1. tt .<.Ksnrirì,E(eitSniaVt"-)-e't2/2dt+.K6n-r/4 , -n S.Kin-ri`+K2n-a/`logn+K3n-(6-ri`)+K4n-'i8=R.(put),. where r=26(o<s<g), == i+43S(t-ss:s-}), ==-}(6>---l;-). Thus, we obtain the following estimation;. Rn=={. '. O(n-6f`logn) (O<SSI/2). O(n-'i8) (6>1/2), completing a proof of Theorem 2.. Acknowledgement. The quthor is very grateful to Prof. K. Yoshihara, Yokohama National University, for his helpful suggestions..

(9) The Rate of Convergence to Normality. 25. References [1] [2]. EssEEN, C. G., Fourier analysis of distribution functions. A mathematical study of the Laplace-Gaussian law. Acta Math., 77 (1945), 1-125. IBRAGiMov, I. A., Some lirnit theorems for stationary processes. Theory Prob. Appl., 7 (1962), 349-382.. ,. [3]. IBRAGiMov, I. A., LiNNiK, Yu. V., Independent and stationary sequences of random variables. Gronigen, Wolters-Noordhoff, 1971.. [4]. OoDAiRA, H., YosHiHARA, K., The law of the iterated logarithm for stationary processes satisfying mixing conditions. K6dai Math. Semi. Rep., 23 (1971), 311-334.. x. [5] YosHiHARA, K., The rate of convergence to normality for absolutely regular processes (submitted)..

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