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(1)Summability of Random Variables Satisfying the Strong Mixing Condition By Ken-ichi YOSHIHARA" 1. Introduction.. 'x,. . Let {6i} be a strictly stationary sequence of random variabies satisfying the strong mixing (s. m.) condition. (1.1) a(n)=AGM9oo' sup IP(AAB)-P(A)P(B)ISO (n--÷oo) BEMee. `. where MZ denotes the o-algebra of events generated by 6., ･･･,6b (aSb). We assume that E16lei<oo and put Eele=:pt.' Let A=:(a.k) be a Teoplitz summation matrix, i.e.. (1.2) /i.m.a.k=O foreveryk, oo (1.3) lim2a.le =1, and niloo k=1 '. oo " (1･4) ･ ,2=,la.le1$M foralln.. -. Here and in what follows, we shall denote by the letter M the absolute constant (not always the same). (Condition (1.3) may be omitted when pt==O). Since. oe oo EZ1ank6le1=21anklE16il5ME16il, -. le=1 k=1 the series 2ka.le6le converges absolutely with probability one. Pruitt (1966) studied the convergence properties of the sequence Zka.leek as n->oo when {ei} is a sequence of independently and identically distributed random variables. The purpose of this paper is to consider the similar convergence properties 11r. ,. of Zleank8k when {6n} is a strictly, s.m. sequence of random variables.. (Theoremsland 2). ' .t. 2. Convergence in probability. In this and the following sections, we assume that {ei} is a strictly stationary, s.m. sequence of random variables with E18il<oo and E6i==pt.. ' Department of Mathematics, Faculty of Engineering, * Yokohama National University. 156,Tokiwadai,Hodogaya,Yokohama,Japan ･ ..

(2) 10 K. YosHIHARA Let F be the common distribution function of the g'les. We adopt the convention that when a.le =O, Ia.kI-i=+oo.. THEoREM 1. Assume thatforsome numbersrand r' (O<r<r') Elgi1"W<oo and Z{a(n)}i/(i"r')<c>o. Further, assume that maxla.kl--->O as n->oo. Then,. le. co. ,:.,ank6le-->pt in Probability.. PRooF. The proof is very similar to the proof of Theorem 1 in Pruitt [2].. First, since Eleil<oo so. (2.1) lim TP(le,E >= T)==O. T.co if iank6kl:$1, (2.2) . rpnk=Ioa"k6lel otherwise. Since m,axla.kl.O, it follows from (2.1) that. 7. '. , '. (2.3) P(:i) a.kekt:l) rpnle) ;!ll ]P(ank6le7Erpnle)=:I]P(l6il> l.1.,1 ). '. Se ￡ 1a.k] SeM h. holds for all n sufficiently large. It will therefore suffice to show that. (2.4) ･ ' 2rp.le-->ptinprobability. Note that. tt. k. tt/. (2.5) E;i]rpnle-pt =]I] a"le{S[xi<i..ki-i XdF(X)mpt}+pt{ li] anlem1}-'O Now, let rp-.le=:rp.k--Erp.k. Then. (2･6) 1nydnklE-{l;2, (2･7) E1rp-nk1;:Ell2Elrpnk1=21ankiE18il, and (2.8) E]rp-.kl"iir5M]a.kli'iirE16,1'"iXr.. Since '. '. li; Slxl<T X2dF. = llk {-T2P(l6ill-li::T)+2SgxP(16l2;x)dx}/ it follows that for all n sufficiently large. ･a.

(3) Summability of Random Variables 1! (2.9) :i] Var ')-7nkS- ]i) lankl2S,.,<,..k,-i X2dF. Se Z la.kl SeM, k and so using (2.6)-(2.8) and Lemma 2.1 in Davydov [1] (2.Io) Var(:i]ijnk)=lI]Var(rp-nle)+2tfl.ll,,t/i.l.,Erp-nj'rp'nk. '. '. co oe i$EM+20,2.=,,i.,[Irp'nHli+i/rllrp-nlelli/t{a(le-1')}ii("r'). '. SM[e+tlli.ll,,tj..,Hrpnjlli+i/r{Elrp.le1}`{cr(fe-]')}i/a+r')] ". SM[e+m,axlank1`tli.ll,la.,･1,20=O,{a(le)}i/(i+r')]. SM[e+max1a.lelt],. i. le .. where ll4U,={E14lP}i/"(P>O) if ElC1P<oo, and t=(1+r)(1+r')/(r'-r). But, by (2.2) and (2.6)-(2.8) E(:i]lrpnle1)2 ;Ill E{]ii) l)7nkl2+2.,IO=O., ,tb.., i)7nj' H ?7nlel}. oo ;:illZE1rpnk12+2Z oo 2 {ElrpnjlEfirpnkl+10IIrpnjHi+!/r{a(le-]')}ii(i'r'} J' le=j'+1 -1. h.. (2.11) . . ･. ;i;M[lI)1anhl+tW.,,tj,.,1anj-1lank1+tfl.i,la.J･1i.I],{af(k)}i'(i'r)]. S M[¥ iank[ +(? 1a.k1)2] S M.. '. Hence, it follows from (2.10) and (2.11) that. Var(Zrp.k)-O as n->oo. le. An application of Chebyshev's inequality completes the proof. bi. ' 3. AImost sure convergence.. ' '. In Theorem 2 (below) we consider the conditions of the following types:. (3.1) Elg,li+i/r<oo,. (3.2) - Znb{cr(n)}C<oo,and ' (3.3) max la. leiS Mo n-d, ･ le. where r, b, c and d are some positive numbers..

(4) 12 K. YosHIHARA THEoREM 2. oo (3.4) ,2=,a.h6leopt a･s･ holds if one of the following requirements is fuly711ed:. (i) (3.1)-(3.3) hold with r>=1, b=O, d>r and c such that. O<c<(d(r+1)-2r2)/d(r+1)2. (ii) (3.1)-(3.3) hold with O<r<1, b=P-1, c==(r+1-2Pr)/(r+1) and d> 2r(1+r")/(r+1), where P is the largest integer such that 2P<1+1/r and r" is an arbitrary Positive number such that Pd>1+2o for some 2o(>O). As in Pruitt [2],.we need three lemmas. LEMMA 1. ILIe (3.1) and (3.3) hold with r<d, then for eve7cy e>O. (3.5) ]Z P(la.leglelle for some le)<oo.. '. n. t. PRooF. The proof is identical to that of [2, Lemma 1] and so is omitted. LEMMA 2. if the conditions in Theorem 2 are satisYied, then. (3･6) :li)P(lank6le1llln"evforatleasttwovaluesofk)<oo,. s. } ,. where O<a<min((d-r,)/(r+1), d(6(r+1)-r)/S(r+1)) and r/(r+1)<6<1-c. PRooF. By the Markov inequality. (3･7) P(la.legkl>=n-a);:Sla.kli'i/rEl&li+i/rna(i+i/r) holds for any a(>O). Hence, from Lemma 2.1 in Davydov [1], (3.3) and (3.7) it follows that for any S(O<5<1) (3.8) P(Ia.leeleIln-a for at least two le) :;l Z P(la.j6jl>m-n-a, Ia.hgkllliinHa). j.tk S M Z [P(1a.j6jl l n-a)P(Ia.k6 le1 >= n'a). j'tk ･. + {P(1 a.S,･1 ). n-a)} O" {cr(k -7')i-6} ]. '. 5 Mi.:.E)k [1ankii'iiria.j･Ii+i/rn21(iti/r). ' ' +{ia.j.1i"i/rna(i+ilr)}5{a(k)}i-'6] '. ' :$M[n-2(cl/7'ar),.2.0,e.,'lah,･1,2co..,1a.le1+nLd(6rLi)'a6rtep.,1a.,･ltpe.,{cu(le)}i-6] '. tt. -,. 'h. s. ,. , $M[n-2(9f.r-ar).±ap-,d(,6r,-i)+a.O.r,Zoo=l{a(k)ii-O]. Whe lkeo e==ljLg-L/hro'ose 6 so that r/(r+i)<6<ii-c. Then, we have that there exists a positive number 2 such that. (3･8) P(lank6fe1>=n-a for at least two le)$Mn-i-2 for any O<a<min((d-r)/(r+1), d(6(r+1)-r)/6(r+1)). Hence, we have (3.6).

(5) SummabilityofRandomVariables . 13 and the proof is completed. LEMMA 3. If the conditions of Theorem 2 are satis/7ed and pt==O, then for. evezy E>O . (3･9) ZnP(iZAanle8lellilE)<OO. ij (3.10) ]ZAank6k==]E){k;iank6ki<n-ct}anfe6le and a is an arbitrary number such that O<cr<r.. PRooF. Let rpa,==(ile; ,'i,,i,a.'fi,k,i<e-a'. a&ddse"tlehi.Entle' if anle==O, then Pnle=pt==O, whiie if a.leto and ({ .3) (with d<r) ". .. ,x. '. ' IP"kl=pt-S:xi<n-atia.pt-iXdF ' ' ;$jixizn-a+dM,IXIdF. Therefore, P.le-->O uniformly in le and 2lea.kP.le->O. Let 4nle=anle(rpAk-Erpale). so that EC.le=:O, EIC.kli"i/r $MIa.kli"'ir and l4.leIS2n-a. Now,. ZAank6le=Zleanferpak=ZleCnk+ZleanlePnk and so for all n suMciently large {1 ZAa. leek1 l-) e} C{12 fe4nk1 >- -S-}. Therefore, it will suffice to show that. (3･11) 2nP(iZle(;nle1i-lliE)<OO･ (a) We prove (3.11> when requirement (i) is fulfi11ed. We note that. (3･12) E{Zkl4nkl}2$M･. F. Infact,foranym , E{,Z'=n,(I4.,l-Elc.,1)2}. ･;･. ft'. '. t ttt. m 5,2=,E(l4nk1-E1Cnkl)2+2,,,.n.s.E(lCnH-E14njl)(lCnkI-E14nfeD. and if r>=1 ' EE((lg.n,iiJ.Eig.℃.C()rf.M,iL"zkici.'1/i)""a(i-i'r'=<-Mn-d'r-aa-i/r)ia.,i. -5Mn.:..all[Cnjl-EICnjllli+iir{a(k-1')}i/(r"i' . ', ' $Mn-ala.,･1{a(k)}i/(r'i) (1';fe),... ･ .....'..

(6) ' 14 ･, K. YosHIHARA '. andsoforsomea'(>O) , , E{,]XM=,(l4nle1-EI4.le1)}2;;lMn'a',2co=,ianle1:;lM･. Therefore, for any m. ' E(,￡M,.,CnleI}2g!$2[E(,ZM..,(1Cnle1mE14nleD)2+(,]M[..),ElCnfe1)2];IEIM , which implies (3.12). We note that if s=-c+1/(r+1), then for any n and k llCnlelli/s$M{n-a(i/S!i-i/r)Ela.krpale1i+i/r}s. SM{n"'a(i/S-i-ilr)max1a.feii+ilr}SsMn-d(i+ilr)s-a'(i/s-i--i/r).. h. So, if r>-.1 and O<cr<r, then for any positive integer m. '. J. E kZM=i4.le i+iir:$[E k2M=ic.le 2](r+i).!2r '. 4 ",. ==[k2M..,ECZk+2,,jl<￡k,.Ec.jc.k](r",i)/2r. ) ,. -<-M[leZM..in-a(i-i/r)[a.h1i+ilr. (3'i3) +,,,.<z?,.LIc.,[l,.,!r"c.klT,/,{cv(fe-i')}c](r"i)i2r. sM[n-a(i-i/r)-"cl/rle2co=11cr.lel. +n-d(i+iir)S-cr(i/S-i-i/r)tW.iia.j.1le]Eoo.,;i{a(le)}yr(r+2)](r+i)/2r. S{ Mn-i-2 where 2 is some positive number which does not depend on m. Hence, (3.11) follows easily from the Markov inequality, (3..12) and (3.13). if requirement (i) is fulfi11ed. ,, (b) Weprove(3.11)whenrequirement(ii)iS'fulfi11ed. Wenotethat ,,,,. .. (3.14) E{ ,2C='O,tc.,t}2PEM. In fact, from Theorem 3 in Yoshihara [3] it follows that for any m E{,ZM=,(14nle1-ElCnk1)}2P;!IilM(,2M=,a;le}P. E{,IEM)=,i4nkl}2Pl:;lM[E{,2M=,(14nle1'E14nk1)}2P+(,ZM=,E14nlel)2P],. and so (3.14) is easily obtained. ･ By the definition of d and P, Pd>1+Ro and so from Theorem 3 in Yoshihara [3] it follows that for anym ,. x.

(7) Summability of Random Variables 15 E tpt.,4.le 2PSM{kZM--,aZk}P. ' (3.ls) $M{m,axla.kl}P{,2oo..,lankl}P. ' E{gMn-Pdf{:Mn-i-Zo. ' ' Hence, (3.11) follows from (3.14) and (3.15). Thus, the proof of Lemma 3 is completed.. PRoOF oF THEoREM 2. By Lemmas 1-3 the proof is obtained analogously to that of Theorem 2 in [2] and so is omitted.. References [1] DAvyDov,Yu.A., Convergenceofdistributionsgeneratedbystationarystochastic. -. i･,N. processes. Theory Probab. Appli. 13, 691-696 (1968).. [2] PRuiTT, W.E., Summability of independent random variables. J. Math. Mech. 15, 769-776 (1966). 'e. K,. x. l･. t. [3] YosHiHARA, K., Moment inequalities for mixing sequences. Kodai Math. J. 1, 316-328 (1978)..

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