9TheScienceReportsoftheKanazawaUniversity，Vol.VI，No.1，pp.9‑14，April，1958
OntheGapSequenceHavingRandomSigns
By
NoboruMATSUYAMA and ShigeruTAKAHASHI(ReceivedJuly15，1957)
1. Introduction
In the present note let f(x)，一 ∞ く xく 十 ∞functionsatisfyingthefollowingconditions:(1.1)

f(x+1)=f(x)，(1.2)

flfwill denote a Borel‑measurableωdx=O，andωdx=1

(1.3)

flfZ

Furtherlet {n，}be alacunarysequenceofpositiveintegers.
The problem whether，fo
ra gapsequence {f(n"x)，} probabi
litylimit theorems
ultifn" doesnot dividenk+l・
holdo
rnot isverydi伍 cInt
hisnoteweconsidera gapsequence{f(n"x) ~，， (t)} ，

where{ヂバ t)}i
sthesystemo
fRademacherfunctionsandprovethefollowingtheorems.
Theorem1. I
ff(x)satisfies，forα>1 andh→0，(1.4)ω‑f(川 )j4dx)1/4=0(1仰 gzjh11")(fljf

thenwe have，for almost allt，N̲

2Jf(n"x)ヂバt)!
Z"mN+ooν2 NlogzNp.
p.m x.

1.

Theorem2. I
ff(x)sati
s
fies

(1.5)

jf(x)j<M

and，forα>0andh→0，0.~f~fω -f(叶的 jdx=0(1仰gjhll") ，

thenwe have，for almost allt，
〆N

¥，・」

ρ9I‑

11x;y/~H :Ef(ω〉 ψω~y I
1= ν2n，:;J.e‑f‑du.
>>"*"";'1¥'Nt=t"'，~"/ J~'-/=JJI:.
，;
1

Wecanprovetheabovetheoremsi
nthesameway asthatof[1].
2. SomeLemmas
Lemma1. Underthehypotheseso
fTheorern1，we have

10N. MATSUYAMA and S.TAKAHASHI

J21ゴヤEf(nhx〉=1，Np.
p.m x.

Proof. By(1.4，) we have

{Jいω ‑1い の ￨2dx)1fLodi￨/ω‑/(川 )4(.同1]4
1=O(1/llog1h1り，ash→O. Henceby(1.3)andthetheoremof P.Erdos[2]，wecanproveLemma1.
Lernrna2. Underthe hypotheseso
fTheorem2，wehave

O(l(n"x)/(njぷIL ""1'" )j，jJ1/(n"x)/(nx)
dx==O(Ik~il"~Ik‑jl"J(Ik‑jl→+∞).

Thislemmai
saspe
c
ialc
aseo
ft
ha
to
fS.lzumi [3].
Lernrna3. (c.f.[4]) Let Z
k，1三五止.
::::;:N ，beasequence of independent randomv
ariablesandE(zρ=0，E(Zk2)く+∞. I
fp andx satisfyMax IZk五三μ，1l:;;
'k:;;
'N

andN

ρxと~E(zわ，thenwehave

仲Pr

3. ProofofTheorem1lnt
hissectionweproveTheorem1.
Let usput，fo
rk=1，2，…
…
，zlI/(x) sk1J，4(f(x)，1Ik(X)=~ 。

(3.
1)

and(3.
2)

ゲ otherwz'se，1，，'(x)=/(x)‑lk(X).

Thenwe have

孟I(x;Ik'(n 〉宇0)1合 I(x;1/'ω￨とがめ 1 O(J11/(x)14dx) 0∞
ρ==Fromt
his，wehave

(3.
3)

p.
p.m x.

~ II/(n"x)1 <+∞，

ByLemma1andthe aboverelations，wehave;EF(nw〉=2JI/2(nkZ〉‑2濯ん(nkx)lk'(仰 〉 +2Jyow〉
N(3.4〉NNN

1=λT+o(λT)+O(NIJ4)+O(1)=N+o(N)，p.
p.inx asN→回.
Nowforeachx，{Ik(匁 kX)'
P，(t)}isasequenceo
findependentrandom variablesandOntheGatSequenceHavz'ng RαndomSigns(3.
5)

HE{jk(nkX)れ(t)}=0，and

手E{(fμ "X) (t))2}=キ/的日〉(3.
6)

仇Furthermorefrom(3.
4) and(3.
6)wehave，foralmost allx，

1
1 会fk2(n"x)

If.N(n.Nx)c
f!.N(t)1<NIμ = o H ‑'r

¥1/2)

，、I ~

f¥logz¥平f"り "X))!

asN →∞. Hence，bytheso‑
calledlawo
ftheiterate
dlogarithm，wehave，foralmost
allx，
N

̲.:
Ef"(n，，x)仇 (t)li・my2NlogzN

N 今回=1，p.
p.tn x.

Sincef"(n，，x) isBorel‑measurable，we have，for almost allt，N̲ :Ef，，(nkx)Cf!
，(t)
(3.
7)

lim .
.
2 一一一一一一
y2NlogzN

p.
p.l
nx.

=1

山 田

By(3.3) and(3.7)，
wecanproveTheorem1.
4. ProofofTheorem2
Inthi
ssec
t
io
nweproveTheorem2
.

弘信

4'e

3︑ノψ'︑
fIb仙

¥jx

ft

n︑F︑
J44

一一x︑︐ノ︑町N

fhSS一一NNZILet usput，forN =1，2・，Forthe proofofTheorem2，iti
ssu偲 cienttoi;
¥howt
hat

ω か f1tzp(予言)dx~e-).'/z ，

(4.1)

p.
p.i
nta
sN →+∞， holdsuniformlyin.
Awhichiscontainedi
nanyfinit
einterval.
Inthefollowing，wemay assumethat(4.2)
Usingtherelation1.
AI孟A
̲2e~= く1+ 心ff+OEim18〉

asx→0，we have

f=J
写(1十ラアヱ乙ψNqf〉

1.
Af(n"x)，"，.
，¥.
‑1!N(必 〉 判 N ぷ〉':"
(1I iCf!
"(t))
ewhereeω=互訪亭f2(n"x)，
JZ .N.Nand口長崎 If(仰 )(x)

"1)
.
N

31N.MATSUYAMA andS.TAKAHASHI

12Thenwe have，by(1.5)andLemma1，l2limc
N(X)
= '~

N‑+∞p.
p.1n X，

血andforallx，limヲN(X)=ON‑+∞

andtheconvergencei
suniform in，t Furtherwe have#(1+ 伊 ωYIi).
/(1S~(l+ザ1)sw ，andA
2
M2.

ICN(X)I孟Hencewe haveiff(1+与手与ω)山ヰJll~(l+空手子Llf!kω)1 2 dXV/2{fl-^'/2-e-~Nω判Nω12 ゐyJ2 =0(1)，uniformly inA andta
sN→+∞.Hencefor(4.1)，iti
ssu
fficientto provet
hat

jidIHR(x，f〉
dxz1

(4.
3)

p.
p.i
nt
，

holdsuniformly in A，where!
!
‑(1x)仇 (t)¥i
A
/
(nk1l
N(X，t)= 平 ll+ν V~T IC'-~./I
fweputI)

fん(t)
= 11l
ρ，
ω‑1，
thenwehave

1f
1l.
lNω1dtf
1fdxイ (1l山，の ‑l)(llNω)‑1)
2=11nf
1
f
dxのf
N(X，t)
万NCY，t)dt‑
1==ff
Mff{1+ ω 47fw〕〉+監型炉型2}dt‑1
向〔f1f
ト A2/併合/(nkYL)ぬの ‑1

=<f1
f
1p
{持/(nkx)/(nkY)}dxdy‑1
回

13OntJt
eGatSeque7Zc
eHaz1uzg Ral
Z
do1Jt Signs=(1(1r).
".
f!
‑.U ，，~， l"(~.._ "'-"-~( ).
"~日JJ.
LN~f( 叩)f(〆 ω，. l"(~，，，_ ，，'¥2ð~'''' J主主~ "l(.̲.~，(r ，，'li'M ̲wh邑re0<ザく1.
By (1.2)，(1.5) and(4.
2)，we haveρ1ρ1:

2dnJNt<ZlIJN(t)2t
三五‑~EtA2M31 1 C J/(川 X)f(lcy))
2ど か どzv.
1
1Onthe other handwe have，by Lemma 2Jkf
f
1(卒ωアd みZkX)f(llf(l1，')SF

2 ~l.!!!..ρ1Z 7 + F 2 7ヨ 斗 仇 心 / 川
where

戸=Min(1，2a).
Ifwe put
ぬ=(が/~) ，then会心/向。)
I"dtく￨∞，uniformly in λ 日記 nce(4.
3) hold
邑 forN = N，， 正ニ 1，2，・Nextwe put，for any fixed忌リ二三九)and1<1・く Nk+t‑Nk，zi=f(7ZNi， +iX) ヂ λTk什 (t).

Then we haveE(z;)ニ 0，.N，， +!-N"~l.N"+1‑]

日 E(Z;2) = 記 f2(1ZkX)，r = Nk斗 1<M2(NH1 -N，，) ニ OO~~-l) ，

孟 1，and，since s
MωIz
る￨亘 A:
f<kji‑z，
1豆 急 く N k+l‑N"

for止ミ止0・ Hencewe have，by Lemma3，k

沼れI>
kli‑り < 2e‑;'，2v

乃'ob.1(for any j¥九 1三玉 N く N"+l一λT!
C.
From the above inequ呂lity，we have

2jProb.( maxた;と kON

12jzil>ki
l‑りl;S;N く.N"+l‑N&.l=210〔よ 1g‑EHK1〉
Thisshows that，forall x，

14

N. MATSUYAMA and S. TAKAHASI
口J.$.N<N，s+l‑N ，;~f(nix)'Pi(t)i=N"十 1<1p.
p.m t.

3k‑?叩1Flinz Max﹁‑一.
l+lh+.
N︐kSince/(1ZkX)is Borel‑measurable andが い 。( V N/c)

ゐ4 叩l$;Nく NII+1‑Nt"

が一一 41linz Max

材一万川甲山4一(4.5)

ρ市

as k→悶， we have，for almost all t，=0p.
p.m x.

By (4.5)，we can prove Th
邑orem 2
固Referenc
記呂[1J R. SalemandA. Zygmund，Some prop町 tieso
ftr
igonometricalseries whose terms haverandoms
igns，ActaMath.，91(195'1)pp. 241~301.
[2J P. Erdos，Onthestronglawo
flarg
邑 numb邑fS，Trans.Amer.Math.Soc，. 67(194型)pp.51~56.
[3J S. Izumi，NotesonFourierAnalysis(XVI)，TdhokuMath. JOUf.，1(1官50)pp. 144~166.
[4J A.N. Kolmogoroff，Uberd
as GezetzdesiteriertenLogarithmus，Math.Ann.，1I:n(1929)pp.
126~135.