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Sci_Rep.Fukushima Univ.No.29(1979) l

Central Limit Theorems for Lacunary Walsh Seriesand an Application

Katsuhiro OHASHI

Branch of Mathematics, F,aa tlty of Et・onomtcs,Fukushima Unit'ersity, Mori-ai l 0- 7,Fuku:shima

In this paper we prove that a central limit theorem for lacunary Walsh series underthe so_called?'Takahashi's gap''and our theorem is the best possible and is applicableto divergent Walsh series.

1 . Introduction.

The purpose of this paper is to prove the following three theorems.Theorem l. Let 、nkl be a seqtience of postti11e intege? ,n t1?? ng

r1?.,/nk> 1十ok (0< c Od".a'・d".1/2) (1_1)and let lakl be a seqtienceof non-negattt)e rea1 mmbers,sati.sfying

Av = (j a i )%_ →十00 and aN= o(ANN °)as N 一→ 十°°. (1_2)1

Then ule ha11e,for any real number x and any set E ⊂[0,11 of Posttitlemeasure

tm ?E 1-'11ωt E :A?i? akω.(ω)d" xl 1= (27r) 'J: e- du. (1.3)

Theorem l js the best possjble jn the sense that under (1.1)the order of the sequenCe iS the best,that is,

Theorem 2. For any g1υon constants 0< c and 0? a ? 1/2,there enst a sequence of Post- flue ,ntegers ln,,l and a sequence of real mm bers la l tohich satisfy (1.1)and

A 1v = (◆a?)?一 十00 and av= 0 (A~,N ')as N 一→ 十°°, (1.4)l

bttt(1。3)ts not true f lor E = [0,1].As an application of the theorem l we have,Theorem 3. Let constants 0 < c and Od":c,d"11/2 and seqttences of Postttυe intege「S

い1k、 and lak1of non_negahve real numberssatlsfy (1.1)and (1.4),resPectit'elyThen the sones

j a- n? di'tl,erges almost etlerytu,here and also ts a Walsh-f louner Sones1

Remark 1 For the trigonometric series,analogous theorems have been obtained in[5.6.7] When E is equal to[0.1],we obtain [1]of A.Foldes.

Remark 2. Using the similar method in cur proof,we can show ,he approximation to a w iener process by random functions constructed of the Walsh series and it is given in[4]_

1

2

hi as h

0

K

CLT for Lacunary Walsh Series and an Application

2.Notations and definitions.

Walsh system 1ωnl is defined as follows;ω。((・))= 1,ωn( ω)= r n,('°)r ? ( ω) - - r n? ((? ),

n=2nl 十2n2十一一十2nk(n1> n2> - - >nh?0),where

rn((・,)一_ r。(2 t,,), r。(ω十1)= r。(ω),r ((,1)= { 1 if (,.)e[0,1/2)- 1 tf (・lt [1/2,1)_The n-th partial sum of Walsh functions called the Walsh-Dirichlet kernel is defined as

D (ω)= j 'ωk(ω). (2.1)

Let lX,,,F;l be a zero-mean square-integrable martingale on a probability space (?,F:P) and let y1=X1,Y =X -X _(n?2).Then to prove the theorem 1,we apply the follow- ing theorem T given by S,Takahashi which seems more useful than any one of D.P_ McLeish in[3].

Theorem T[9]. Suppose that there extst a seqttence lC.l of positive nttmbers f or ωhich 11m C,,= 十00,and a random υariable Z((,1'),satlsfyirtg

(L- I) for any given E> 0, 11m C-n21:: E IY?1(1y??>:' o n)l = 0,l

(L- II) ltyvj C-,,2j.r YZ= Z;. m Probabi lity.

Then f or any set F eσ( 0 .F )and any real number :,c(x,,0)1

?? Pl(・,e F : X.(ω)/o nd" - 2Tr )-''f F、(「 : e- du)dp ,

ulhere(;( OF )dertotes the σ一algebra generated by the algebra OFn and x;/0 1.s十00(or- 00) zf .x1is positive (or negati11e),

3.Proof of Theorems.

Let ? 0)=0,P(k)= n,lax lf :n1<2kl (k= 1,2,- - ).

When n_ 1> nplk,,1;we have that2kd".n <n .<--<n <2-pl k 1+ 1 ? ki , 2 ? k' 11

and by (1.1)

2? np_ /n _ ? 1十c?? :j -'? 1十一 一 )- ? k,)1? - -'.

Thus p(k十1)-p(k)d"lc-'p(k十1)十1d" C? k十1)'. Further by a < 1/2,? k)/p(k十1) t as k ,00,hence

? k十1)- ? k )= 0 (? k )°)as k - 00 (3.1)Now let

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Sci_Rep_Fukushima Univ.No.29(19791 9

△f(ω)= ?j '1akωn?(ω) (f'= 012,- - ). (3_2)ip l 1-l-1

Then by a v=o(ANN-°)and (3.1),

1△,(t,J)l< 1:j 111a kl = 0(A _ ). (3・3)P i l l

For any natural number N there exists a natural number M such that ?M)< N≦? M十1)and letting B = Ap- 1 (j=0,1,2,-- ).

A?'? a,,u k(ω)= (BM_,/A v).B -M'1M? '△1(ω)十A -? ?? 1,a ωn,,(ω)

= (BM_1/AN)11(.tv)十12(? ,Say・

For 12(N) by (3.3)

A ?1( j la kωn- l)? A NN1 i la k、= 0(B M/A .~')・? ? 1+ l ; Mt' l

By (1.1)we see easily that BM/A?, ,・l and BM_,/A.? '1aSN '°°・Hence we have 1(N)一→0 as N 一→00,For I,(tv),let F =o(ro,γ,,- - 「,,_,)(n= 1;2,- - ),x = j ΔJ Ω=[01],F= the Beret field of [0,1]and P= the Lebesgue measure in [0,1]・Then jx ,,,F l is mean_zero squre_integrable martingale on the probability Space(?;FP)

and F=a(? Fn) To obtain the result of theorem 1, we must show that 11(N) Satisfiesconditions of theorem T.But by (3。3),for sufficiently large n,1(1Δkl> ,B,,)=c it n1d" kd"nfor some n,Therefore when Cn=Bn,(L-I)holds under our condition.To See that (L-ll)1s va1jd for z=1;jt js evident from the fo11owing lemma 1.This completeS the P「cotof the theorem 1.

Lemma l [8].

?? ? l iB:1? △?(ω)_ 112dtl1_0.

proof of Theorem 2 We need the lemma 2and3,which fo11ows.Let lXnl be a Sequenceof independent random variables on a probability space (a,F P) and E X・= 0,E X2=σ2<十。。(n=1,2,_ _ )and s2= j σ?.Firstly we denote J.W_Lindeberg-W.Fe11er thee「erm[2,P.292] Theorem L_F Let s? ,十ool ando.,=o(s?)as rn ,°°.Then for any 「eat X,

1imP IS l j Xkd"・X、一 一 1

if and only if,for any ,>0,

11m s一前21_ E IX?1(1X,,1> E;s,,,)l = 0.m→a i

From the theorem L_F,we have at once the following lemma 2.Lemma 2. Under s,. ,+00 and σ,,,= o(s,,,)as m一→°°,if (3,4)holds,

(3.4)

(3.5)

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I O K.0hashi :CLT for Lacunary Walsh Series and an Application

for any ,:> 0, lim .1:: P 11Xk > ?sml = 0__ a t

Lastly,we describe the estimate of D,as the lemma 3.Lemma 3. Ror any natural numbers ? and m

11ω, [0,1] :11) (2'''ω)l> ?/31?≧1/ ? _

Now let us prove the theorem 2,Let 0< c and 0< ad" 1/2 be constants given in theorem 2.Let us put

l ・ 1

{r l a :[M - )_?f・)_,)Since P(j 十1)- ? j )- a-1J '-'1・ and P(j )°?*''j as f 一 十°°,

e(f)一β(a)j as f 一→ °°, (3_5)

where β(a)= {1/c if 0< a< 1/2min(2,1/c) if c,=1/2.Next we put n1= 1,n- = [nk(1十ck-')十1] (k十1くP(j。)) and when n,,f1(j。d" f) is defined,we put

n - = {- 1+ ? ) if l d"1? d";?(j )[nM_ _1(1十CMf )-a)十1] if 1(f )< 1 < P(.f十1)- P(J1).Further for j,d" f let us put n?l,1=2?l1,Where

q(j )= mm lm :2n'/n,ｻ1f1_1> 1十c(p(j )- 1)-°l if j d" fThen the sequence ln?1satisfies n- /nk> 1十ck-' (k=1,2,-- ).Fer ia,,l ,we put

ak= {1 if ? j)d" kd" d "

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Sci_Rep Fukushima Univ.,No.29 (1979) 11

we recognize that lx Jl is a sequence of independent random variables,becauSe RedemaChe「 functions lr.l are independent random variables.Let

/ X (ω)dω= 0,σ-= f X (ω)d(f., = 一 十1 (f= joi e十1,- - ),

and s:= .i o?= j二(a(J)十1)- (β(a・)/2)m' as m - °°・

Then by (3.5), σm- (β(c,)) m'1 andsm- (β(a)/2)1m as m- °°・Hence σm=o(sm)and by (3.6),

(3.7)

since x (?)__ D~ ,(2?f1ω)_ 1,the lemma 3 and sm< ?(m)for large m imply

l基ji l lωt[0,1]:一 ω)l>(β(a,)%/4)Sml l≧Cl? 11/J≧Cl,>0. (3・8)

For A?'SN(ω),]et's suppose that (1.3)is valid for E=[0,1],that is,, , ・ _f

l i m i tω e [0 ,1 ] : A ? : ,,. , ,m,S ? ,,,,+ , , ,, ,,? ) ≦ x i i = ( 2 7r) '' f e ' d u

Considering that (3.7)and sm-A?,1・e1w,,

?? 11ωe[0;1]:s-m'? x ,((fJ)≦xi i = (2Tr)-% j -' du

When we put e=(β(c,)'/4 in the lemma 2,

1jm j 11ωe[0,1]:lX 1((,,)1> (β(a)'/4)s,n il = 0.一 ? lm 21

This ls contrary to,(3.8)and the case a,=01s known.Hence it completes the P「oct et the of the theorem 2_R・cot of Theorem 3 To show the result,we can supposetheboundednesSOf the Sequencela 1.By easy calculation we have

j ai /Ak- 1ogA.. as N→ °°.

Thus,let bk=_a?/Akand B? = i b and we have

k bk/Bk- k 'b?/(log Aj)'' = 0 (1/(log A?)%)= o(1)aS k - °°・since B _ →。oas n _ →00,by(1.4),then the theorem l implies,for any set EC[0,1]With lEi> 0 and any real r,

?? lE i-'11ω.eE :B? '? b?一 ω)≦:1;・l l = (27r)-% /:e - du

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12 K.0hashi :CLT for Lacunary Walsh Series and an Application. ~ '

Hence when we put TN(、t,,)=◆bkulnk(?)(N=1,2,一一),for any sequence of real numbers1

1M satisfying ljm Nk= 十00,

??? lTNk(o?)1= 十°°,a.e_On[0,1]. (3.9)

Now we denote the M partial sum of the seriesj aku1? (ω)by S,v(a,)= j akωn,,(ω).

If there exists a set E??[0,1]with fEel> 0 such that ? akωnk(ω) converges a.e. on E。,then by the well-known Egoroff's theorem,there exists a set E,⊂E。with lEi,1>0 such

that S,v(,・))converges uniformly on E1.Then from the relation that

T N(ω)= A -N'S ~,(?_,)十j 1(1/A - f l Akャ,) Sk(ω),

we have the result that 7;v(ω)converges on E,.However,this is contrary to (3.9).Thus,

? akω?(ω)diverges almost everywhere.Next let;s suppose that ◆a ωn?(,L,)is a Walsh-Fourier series.By[10]of C.Watari,the series converges in If -nom for e< p< 1. Hencesome partial sums S?(ω)converges almost everywhere.However this is contrary to the

above result.Thus ? a ωk(a,)is not a Walsh-Fourier series.This completes the proof of the theorem 3.

Remark 3. In the theorem 3,we can replace an= 0(Ann一つbyan= 0 (An(log An loglog An- - log,,An)1イ'n -a),p≧2、

Acknowledgement.

A part of the results in this paper was obtained during my stay at Kanazawa University in 1978.I would like to thank Pr・ofessor S.Takahashi for his help and encouragement.

References

[1] A_Foldes :Central limit theorems for weakly lacunary series,Studia Sci.MathHung,10(1975)141-146.

[2] M.Loeve Probability theory I,Spriger - Verlag,19774-th edition.[3] D.L_McLeish Dependent central limit theorems and invariance principle,The

Annals of Probability,2(1974)620-628_[4] K.0hashi :A note on the functonal central limit theorem for lacunary Walsh

series,Sci_Rep.Kanazawa Univ_,23(1978)65- 68.[5] S.Takahashi :On lacunary trigonometric series,Proc.Japan Acad。, 41 (1965)

503-506_[6] S.Takahashi:On the lacunary Fourier series,Tohoku Math.Journ.,19(1969)79_ 85.[7] S.Takahashi:On lacunary trigonometric series II Proc_Japan Acad.,44(1968)766_ 770,

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Sci_Rep.Fukushima Univ.,No.29(1979) 13

[8] s Takahashi :A statistical porperty of the Walsh functions,Studia Sci.Hung.,10(1975)93-98.

[9] s Takahashj :A version of the central limit theorem for martingales P「OC・Japan Acad.,55 SerA (1979)163- 166.

[10重 cwatari :Mean convergence of Walsh Fourier series,Tohoku Math.JOum.16(1964)183-188.