Statistical Independence and Normal Numbers: An Aftermath to

Transcription

Statistical Independence and Normal Numbers: An Aftermath to Mark Kac's Carus Monograph
Author(s): Gerald S. Goodman
Source: The American Mathematical Monthly, Vol. 106, No. 2 (Feb., 1999), pp. 112-126
Published by: Mathematical Association of America
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StatisticalIndependenceand Normal
Numbers:An Aftermath
to Mark Kac's
Carus Monograph
Gerald S. Goodman
1. INTRODUCTION. In 1958, Mark Kac deliveredthe prestigiousPhilips Lectures at HaverfordCollege, whichformedthe basis forhis Carus Monographon
StatisticalIndependence [2]. As a student,I had the privilegeof attendingthose
lectures.By foragingin theiraftermath,
meaningliterally,"that whichgrowsafter
the harvest,"I have been led to the followingideas. They may be regarded as
comprisinga long footnote,or possiblya missingchapter,of that Monograph,
composed in memoryof its authorby one of his erstwhile"guinea pigs."
In [2], Kac presentsan arithmeticalmodel for coin-tossing,due to Borel [1],
whichculminatesin an analyticalproofof the StrongLaw of Large Numbersfor
Bernoullitrials.The idea is to identify
the digitsoccurringin the binaryexpansion
of a point w E Q = [0, 1),
Ea
2'
'
(1.1)
j=l
withthe outcomesof a faircoin-tossing
experiment.The j-th digit,aj(w), equals 1
if tails occurson the j-th toss,and equals 0 if the outcomeis heads. Then
Sn( w) =
a,( w) + a2( w) + -- +an( w)
(1.2)
is the total numberof tails in the firstn tosses,and Borel's StrongLaw of Large
Numbersassertsthat,when fQis endowedwithLebesgue measure,
1
Sn(w&)
- -> as n -> oo.
n
2 a.e.,
(1.3)
Kac's proofof Borel's theoremmakes use of the Rademacherfunctions.These
EE N and w E fQ,as
are defined,
foreach j
rj( w) = 1- 2aj( w),
(1.4)
and theyrepresentthe net gain of a gamblerwho bets one florinon the outcome
heads at the j-th toss.The orthonormality
of the Rademacherfunctionson the unit
intervalfQ,and, indeed, of theirfiniteproducts,is a consequence of the factthat
the binarydigitsare statistically
that Kac
independent.It is this orthonormality
exploitsto give a simpleproofof the Law of Large Numbers.
Kac goes on to interpretthisresultin termsof the existenceof simplynormal
numbers.For such a number,the relativefrequencyof the l's, among the firstn
digitsin the binaryexpansion of its fractionalpart, tends to 1/2 as n -> oo.It
followsfromBorel's theoremthat almostall real numbershave thisproperty.Kac
pointsout thata similarresultholds forany integralbase b > 1, sketchesa proof,
and concludes,withBorel,thatalmostall numbersare simplynormalto everysuch
base.
112
STATISTICAL INDEPENDENCE AND NORMAL NUMBERS
[February
Now, it is known[8] thatanynumberthatis simplynormalto all bases thatare
positivepowers of an integralbase b has the followingproperty:Everystringof
finitely
manyb-adicdigitsoccursin the base b expansionof its fractionalpartwith
asymptoticrelativefrequencyl/bm, where m is the lengthof the string.Partial
overlaps of the stringwith itselfare counted as distinctoccurrences.Numbers
exhibiting
thispropertyare termednormalto base b, and the factthatalmostevery
real numberhas thispropertyis knownas TheNormalNumberTheoremforb-adic
digits.
Since Kac's use of orthonormality
leads to such a simple proof of the case
m = 1, b = 2, it is naturalto ask whetherhis method can be adapted to give a
directproofof the NormalNumberTheorem,at least forstringsof binarydigits.It
turnsout thatthiscan be done by replacingRademacherfunctionsby theirfinite
products,which themselvesform an orthonormalsystemknown as the Walsh
fuinctions.
Kac presentsthemin an exercise,withoutany mentionof theirpossible
connectionwithnormalnumbers[2, p. 11].
That discoverywas made by Mendes-France [6], who showed how base 2
normality
can be characterizedin termsof the Walsh functions.He thenused this
characterizationto prove the Normal Number Theorem for binarydigits. His
approach, which, as he showed, generalizes to arbitraryintegralbases b > 1,
makes use of Haar measure,group characters,a generalizedWeyl Criterionfor
asymptotically
equidistributed
sequences,and Birkhoff's
ErgodicTheorem,applied
to the dyadicmap.
We shall show, instead,how the orthonormality
of the Walsh functionsleads
directlyto a simple proof,a la Kac, of the Normal NumberTheorem forbinary
digits.The only additional idea required is the well-knownobservation,due to
Wall [10] and used by Mendes-France,that a numberis normalto base 2 if and
only if the iterates,under the dyadic map, of its fractionalpart are uniformly
distributedin the unit interval.Using this idea, our proofproceeds in the same
spiritas one suggestedby Kac-again, in the formof an exercise-for proving-the
classical theoremof Weyl [11] on the equidistributionof the fractionalparts of
multiplesof irrationalnumbers[2, p. 41].
We can extendthisapproach to integralbases b > 1 by defining,
withMendesFrance [6], the b-adicRademacherfunctionsas
) forall w E Q, I R
b
E
r1( )=exp2
of the point w,
wherethe bj are the b-adic coefficients
v0 bj()
bJ1
and definingthe b-adic Walsh functionsas theirpower products.When b = 2,
of rj agreeswiththe
bE {0, 1} and Euler's formulaensuresthatthe new definition
old one.
These b-adic Rademacher functionshave mean zero, and all that is needed to
establishtheirorthogonality
(over the complexfield) and furthermultiplicativity
propertiesis the formula
k
Hrexp(itLjbj(
__1
())dw
=
k
fH
j=10
exp(i,tjbj( w)) dw,
values of k EE N. But
forsuitablevalues of the real parameters,Ujand arbitrary
the validityof such a formulais guaranteedby the statisticalindependenceof the
1999]
113
as Kac well knew [3], and its proofis virtuallythe same as one foundin the
openingpages of [2].
In Section 8, we givethe details,and show how the reasoningused to establish
the base 2 normalityof almost everyreal numbergoes throughin the case of
generalintegralbases b. Once done, it remainsonlyto collectthe exceptionalsets
in orderto arriveat a new proofof The NormalNumberTheorem,in itsfullforce:
almosteveryreal numberis normalto everybase.
We then go on to examine more carefullythe connectionbetween the multiplicativityof the b-adic Rademacher functions,understoodas the vanishingof
theirmixedmoments,and the statisticalindependenceof the b-adic coefficients.
Using a device of Renyi's[9] we are able to draw a remarkableconclusion-the
propertycan be
two notions are entirelyequivalent! Since the multiplicativity
establishedeasilyby elementaryanalysis,thisyieldsa new proofof the independence of the b-adic coefficients.
The same ideas can be applied to b-adic Walsh functions.While theythemindependent,we findthatthe ones whose indicesforma
selves are not statistically
geometricalprogressionmade up of fixed multiplesof powers of b are. The
statisticalindependence of the b-adic Rademacher functions,and thus of the
followsas a special case.
b-adic coefficients,
bj's,
2. THE RADEMACHER FUNCTIONS. The functionsrj are defined by (1.4).
Since the binarycoefficientsai satisfy
~~~1
1
jaj(w)
do) = -for j EJ,
frj( )
dw = 0 forj E N.
it followsthat
The statisticalindependenceof the a1 impliesthatthe rj are also independent,
Formula
and, since theyeach have mean zero, theysatisfythe Multiplicativity
f1rjl(
(l)
&w)rj2(
...
rik( @)
dw = 0, k EEN,
(2.1)
wheneverthe subscriptsare distinct.Since rf2= 1 foreveryj, it followsfrom(2.1)
systemon fQ,thatis,
thatthe Rademacherfunctionsforman orthonormal
f1rj( w)rk( w) dw
=8jk
forj, k e
.
(2.2)
3. KAC'S PROOF [2], [3]. In viewof (1.4), the limitingrelation(1.3) is equivalent
to the assertionthat
Rn(o)
-Oa.e., as n
oo,
(3.1)
where
Rn( w)
=
rl( w) + r2( w) +
?+rn(w)
(3.2)
is the difference
betweenthe numberof heads and the numberof tails in the first
n tosses.
114
[February
A directcomputationusing(2.1) and (2.2) showsthat
= n + 3n(n - 1).
f[Rn()]d,=n+2!2!d(2)
Consequently,
fl[ Rn(W) ] d
E
<
(3.3)
00
and it followsfromBeppo Levi's Theorem that
4-'[Rn~w)
n=l
]4d w
<
oo a.e.
This impliesthat
[R~(w)
->0 a.e., as n
]4
-> oo,
whichis clearlyequivalentto (3.1).
4. A VARIANT OF KAC'S PROOF. The precedingproofuses (2.1) up to fourfold
of
products,whichis evidentlya strongerpropertythan the mere orthonormality
functions
expressed
by
(2.2).
However,
a proofof (3.1) thatuses
the Rademacher
of the rj (along with the uniformboundedness of their
only the orthonormality
absolute values) can be based upon an argumentemployedby H. Weyl [11] in a
similarcontext.
(2.2) impliesthat,foreach n E ,
The orthonormality
1
[Rn( W)]2 dw = n.
It followsthat
fl[Rn2(W) ]2
E
du < oo,
O
n=1
and, therefore,reasoningas before,
2
n
> 0 a.e., asn->
(4.1)
c.
Now, to each value of n, not a perfectsquare, thereis a unique positiveinteger
mn such that m2 < n < (ma + 1)2. Clearly, m '
and the factthat Ir,
=
1 forall j,
oo as n
->
oo. In view of (3.2)
+ rM2
+ Irnl< IRm21+ 2mn.
+11+
IRnI < IRm21
Dividingby m2 and using(4.1) with n2 replaced by m2 yields
IRn(
mn
w,)I
2
-Oa.e.,asn
->oo.
Since m2 < n, (3.1) follows.
5. SHIFT DYNAMICS. Define the binaryshift(or dyadicmap) T on fQ by the
formula[2, p. 19 and p. 93]
(5.1)
T( w) = 2 w (mod 1).
1999]
115
Then, when a is expressedin termsof its binaryexpansion(1.1), T(w) takes the
form
T(wa)
=
Z
2'
L
_a
_
j=l1j=l
Makingthe conventionthatbinaryrationalshave finiteexpansions,the uniqueness
of the binarycoefficients
gives aj1+( w) = aj o T( w) forj E N. Iterationyields
aj+k(w) = aj o Tk( W) forjE
N, k E No,
(5.2)
and, in particular,
al+k(wt) =a, o Tk()
fork E N0.
(5.3)
Now, a, is the indicatorfunctionof the interval[2, 1). In view of (5.3) and the
definition(1.2) of Sn,
+a, o Tn-1
S,j( ) = a1( ) + a1 o T( )) +
Thus, Sn( W) countsthe numberof timesthatthe orbitof w under T, thatis,
,T(Z)..
Tk(@)..
is foundin [2,1) duringthe firstn steps,startingfromstep 0. The StrongLaw of
Large Numberscan thereforebe interpretedas sayingthat therelativetimethatthe
orbitof w occupies[1/2,1) tendsto 1/2 as n -> oo,for almost all startingpoints
EE fQ,in accordancewith[2, p. 93].
of the Law of Large Numbersleads to a formulaThis dynamicalinterpretation
tion of the NormalNumberTheorem in an analogous way. Let
al,,
2
*....
a*
(5.4)
n
denote a stringof binarydigitsof lengthm. Set
m aj
I= E
j=l
~2-m*
and let
II [=
) *(5.5)
2m
n S2m
Then I, is the l-th binaryintervalof orderm, and it consistsof those pointsin fQ
whose binaryexpansionstartsout with (5.4). With this enumeration,the lexicographicorderof the strings(5.4) is expressedby the naturalorderof the intervals
(5.5), in termsof increasingvalues of 1.
If we now denote by XI, the indicatorfunctionof the intervalII,
1
-
n-i
E XI!o T(k)
nk=
is the average numberof timesthe string(5.4) occurs among the firstn + m - 1
of w,whereoverlappingscountas multipleoccurrences,and an
binarycoefficients
occurrenceis marked at the momentwhen the stringstartsto appear. Any real
numberwhose fractionalpart is a will thenbe normalif,foreach m E ,
in-1
-E
k=O
1
XI,Tk(w)'>
212
, as n
,oo
(5.6)
where XII is the indicatorfunctionof a genericbinaryintervalI, of order m, and
be a conse1/2m is its length.The Normal NumberTheorem will, accordingly,
quence of the assertionthat(5.6) holds a.e. on fQforeach such binaryintervalI,.
6. THE WALSH FUNCTIONS. Kac introducesthe Walsh functions,as follows.
EE R, let
For each j
2j = 2i1 + 212
(6.1)
.+2is
+
< j] are in R, be the unique expansionof the integer2j in
where il < j2 <...
base 2. Then, foreach j EE N, definethe Walshfunctionsby means of the formula
&j(w)
=
rjl( w)rj2(
w) ...
forw E
rj(w)
fQ,
(6.2)
and set w0- 1. The Walsh functionsthus constitutean enumerationof all the
finiteproductsof Rademacherfunctions.
The Walsh functions,like the Rademacher functions,have a probabilistic
in termsof coin tossing.For each j EE R, wj representsthe net gain
interpretation
of the gamblerwho bets one florinon the outcome thatthe total numberof tails
, ]s is even.
occurringon the tosses 11, ,
Because of (2.1), the Walsh functionsare orthogonalon fQ,while the identity
lw21 -1 implies that they are orthonormal.For each mEE N the 2' functions
W0,W1,... I,W21 are constanton binaryintervalsof order m. The range of any
such functionthus correspondsto a vector whose l-th componentis the value
of the functionsmakes
takenby the functionon the intervalI,. The orthogonality
these vectorsorthogonaland thereforelinearlyindependent.It followsthat the
functionsthemselvesare linearly independent. Consequently,any real-valued
functionf thatis constanton binaryintervalsof orderm can be writtenas a linear
combinationof the first2' Walsh functions:
2n
f(f)
- 1
j=0
((t) , @
Ajwj
E
Q
(6.3)
of f:
coefficients
wherethe weightsAi are the Fourier-Walsh
Aj = |f0)f( ) ij(
d@, j = O,..., 2m - 1.
In particular,the functionXI, can be writtenin such a form.In thiscase,
Ao =
1
1
XI,((w)wO(w)dw = 2m
(6.4)
while
Aj =
J1lx11(w)w1(w)dw =
1
?-m
]=
1,
,2'
-
1,
where the value of the + signis such as to make sgn[Ajw] = + 1 on I,.
7. PROOF OF THE NORMAL NUMBER THEOREM FOR BINARY DIGITS. In
order to prove that (5.6) holds a.e. for any binaryintervalof the form(5.5) with
m ERN arbitrary,
observethat(1.4) and (5.2) implythat
rj+k( w) = rj o Tk( w)
1999]
forallw
E
Q, j
E
, k E No,
(7.1)
117
so that,in view of (6.1) and (6.2),
W2kj(w)= wj]oT ( w) forall
EE f,
j, kE No.
(7.2)
It is easy to verifythat,for each j EE R, the subset Wj, w2j, ... , W2kj. . . . formsa
multiplicative,orthonormalsystem,so the reasoning of the previous sections,
applied to the sums wj(W) + w2j(W) + .+W2`1-i(w),
insteadof to the R", shows
that,foreach fixedj EE F,
1 n-i
n- E wjoTk(W)
-> O a.e. as n>
k=O
,
while,trivially,
1 n-i
>1asn
EwooTk(w)
n k=O
-
>oo.
Consequently,using(6.3) withf replaced by XI,,
1 n-1
-
E
nk=O
1 n-1
x11oTk()
=-
n
k=O
2in=
j=0
2m-1
A1woP()
j=O
k
l n-1
Ai _ , wjo Tk(
n
oJ
k=
)
AOa.e. as n >oo.
Since, by (6.4), AO= 1/2m,(5.6) holds a.e., and the theoremis proved.
8. GENERALIZATION TO b-ADIC DIGITS. Suppose that b > 1 is any integral
base. Let the b-adic expansionof a genericpoint w E fl be
g
i
b1(wo)
(8.1)
1
with the conventionthat b-adic rationals have finiteexpansions.Following [6],
definethe b-adicRademacherfunctions
r (w) = exp
forall w E Q, j
(
EE N.
(8.2)
Thus, the rj assumevalues in the cyclotomicgroupof b-throotsof unity,and each
root is takenon a subsetof fl havingmeasure 1/b.
It followsat once that,foreach j EE N and any integerd,
dw=O,
frjd(w)
unless d
(8.3)
0 mod b. It is also evidentthat,when d is in the rangefrom0 to b - 1,
rd( ()
= rb-d( w))
forall
E fQ,j E
,
so the complexconjugatesof powersof the rj are expressibleas positivepowersof
(8.3) holds also when rj is replaced by its complexconjugate,as
rj. Accordingly,
could have been seen directly.
As statedin the Introduction,
we shall make use of the formula
1j=i
|
1
k
Hlexp(itub;( cw))dw =
~~~~~~k
H flexp(i,ubjw(w)) dw
]=i
0
(8.4)
k EE N. The left-hand
forcertainvaluesof the real parameters
,Ujand arbitrary
function(Fourier
characteristic
integralis one formof the multidimensional
considered
as randomvariableson fQ
ofthefirstk b-adiccoefficients,
transform)
thatassurestheequalityof the
independence
[2, p. 42],and it is theirstatistical
proofcan be based uponKac's demonstration
of
An elementary
twoexpressions.
[2,p. 7].
an analogousformula
Indeed,letP denoteLebesguemeasureon f. Then
f H exp(i,ujbj(c))
k
k
E
=
dcv
= 1 **....bk(c
H| exp(i,8j,3)P[{b1(c)
1,XPk
j=1
)
1=0}
overtheset0,1,... , b - 1. Usingthestatistical
wheretheP3irangeindependently
becomes
ofthebj's,thelastexpression
independence
k
E
i. .
k
Hl exp(i,uj,813)HP{bJ(
c)
.k
J=
j=1
k
= 83j}=
I
Hexp(i,uj1P)P{bJ(c)
Pk =1
1 .
=
18j,
which,in turn,reducesto
k
n
b-1
cP b38} )
Hl E exp(i Pj8)P{bj(
j=1
=
p1j=o
Hrj=
f1exp(i,jbj (co)) dcv,
] =10
as required.
between0 and b - 1,
Settingyj = 2irdj/bin (8.4),wherethe dj are integers
Formula
equation(8.2),theMultiplicativity
yields,inviewof(8.3) and thedefining
functions,
fortheb-adicRademacher
f'r'd(c
drkk()
d)r'2(cv)
dc =
Oforall k E ,
(8.5)
hasthevalue1. Recallingour
unlessall ofthed'svanish,inwhichcase theintegral
to holdwhenany
we see that(8.5) continues
comments
aboutcomplexconjugates,
Whenb = 2,
are replacedbytheirconjugates.
in theintegrand
numberoffactors
chosenand the
in thepresentform,
whenk is suitably
(2.1) can also be expressed
all
are setequal to 1,whiletheotherexponents
oftheselectedfactors
exponents
vanish.
takeanyj E N and makethepartition
To generalizetheWalshfunctions,
bj = dj1bil + d12bi2+
*..
(8.6)
+dj1bis,
in therange1,. . ., b - 1. Thenset wo 1 and define
wherethe d's are integers
theb-adicWalshfunctions
wj(wo)) =
rdi,u(
)rf'i2()
r'
rdis(w),
cv E Ql, j
E
N.
(8.7)
[6,p. 44].
Thisdefinition
agreeswiththatofMendes-France
thattheb-adicWalsh
By(8.2) and(8.7), 1wj12= 1 forall ] E N0.It thenfollows
functions
theorthonormality
relations
satisfy
f1Wj(cv)cvk(c)dcv=
8jk
forj,k E N
?
(8.8)
for,by (8.5) and (8.7), thevalue of the integralcan be 1 onlyif wj and Wk are
and theircorresponding
productsof the same b-adic Rademacherfunctions,
powersagree.
1999]
119
Instead of the dyadicmap (5.1), considerthe b-adic map T(w) = bw (mod 1)
and note that the uniqueness of the expansion (8.1) implies that bj+k(&w)
bj? Tk(w) forj EE R, k EE No.Thus, in view of (8.2), we also have
rj o Tk(
=
rj+k( (a)
(a)
forall wEE
, k E No,
f, j E
the same as (7.1). Consequently,by (8.7),
whichis formally
wjoTk(
&J)
=
rdl+Joj) r
J2 +
... r
&)
(8.9)fl
(fik
( &) I
and so, in view of (8.7) and (8.6),
Wbkj( w)
= wjo Tk( w) forall
wEE
f, j, k E No,
(8.10)
whichgeneralizes(7.2).
Let m be any naturalnumber,and take any string
( 8.11)
1311 1321 -* *,* O m
of m b-adic digits.Let I denote the b-adic intervalof order m made up of those
points in fQwhose b-adic expansionstartsout with(8.11). Clearly,I has length
1/bm.
In view of (8.8), the firstbm b-adic Walsh functions,wO,w1,... , Wbm-l, are
linearlyindependentover the complex numbers,and their span consists of all
complex-valuedfunctionsthat are constanton b-adic intervalsof order m. In
particular,XI can be writtenas
bi-1
X,(a) =
E
j=o
AjWj(w),
(E fQ,
(8.12)
wherethe weightsAi are givenby the formulas
XI( )wj( (D)d Q)= bm, j = 1, . .. I bm_1
Ai
Here, -qjis a b-throot of unityforeach j, and
i*1
J X.( w)wo( w) di
Ao=
~~~~~1
= bm*
(8.13)
It follows,again from(8.8), that,foreach j (E R, the functions
(8.14)
W; , Wb ,I... I,Wbkj , . . .
themselvesforman orthonormalsystemon fQ.Accordingly,
in view of (8.10), we
can establishthat,foreach fixedj EE I,
1
-
n-1
n k=O
wj o Tk( w)
a.e. as n
->0
(8.15)
-0,
by appealing to the reasoning of Section 4, provided we replace the second
momentsof the Rn(W) thereby the second momentsof the absolutevaluesof the
sums
Wj(W) + Wb](W) +
forj = 1, . . . , bf+1
foreach k E N,
and n
EE N.
Wbkk (k( )
120
STATISTICAL
=
...
+Wbf-l(W)
A more involvedcalculation,usingthe fact that,
+k((J)
INDEPENDENCE
j2 .+k())
r
k
)
AND NORMAL NUMBERS
[February
whichfollowsfrom(8.9) and (8.10), and using,mutatismutandis,the Multiplicativand that
ityFormula (8.5), showsthatthe functions(8.14) are multiplicative
f1W(
Wbl-lj(w)4 dw
i) + Wbj( W) +
< n + 3n(n - 1),
so Kac's originalargumentof Section 3 can stillbe used.
Once again,
1
n-i
)
EwOoTk(
-
k=0
>oo.
lasn
Consequently,because of (8.12) and (8.15),
1 n-i
EXioTk(w)X
n k=0
AOa.e.,asn
-
>oo,
just as in Section7. But, in view of (8.13), Ao= 1/bm.This provesthatthe relative
frequencywith which the string(8.11) occurs in the b-adic expansion of the
fractionalpart of almost everyreal numbertends to l/bmas n -> oo,and that is
enoughto establishthe NormalNumberTheorem.
9. INDEPENDENCE AND MULTIPLICATIVITY IN BASE 2. Returningnow to
base 2, observethat
1 + yN(9.1)
81/71
2
whenevery, q
=
?1, as noted by Renyi [9, p. 130]. Consequently,forany j
1 + y1r1
2
=Arj=
E
(9.2)
yj},
where rj is the j-th Rademacherfunctionand yj = + 1. Since
m
17x{1=y,j}
=
J=1
X{rl=y ,. .r, =,=yI1}
it followsthatforany m E NJ
Xrl=y
r (=iy
,
)
=
(9.3)
Renyiuses thisformulain the followingway.He expandsthe productin (9.3) to
get
jj~1Yi
ri)
/rl~~~~~
( 2 )
=
i(+
2(
r+
Y
j
94
+ -+ jH yjrj . (9 .4)
yj^ylrjr,
Setting
(9.5)
yj 1=1-2a, for =1, ...,m, whereaj = O orl,
oftheindicator
function
thengives,inviewof(1.2) and(9.3),an expansion
XI,ofa
2mWalshfunctions,
as
(5.5) oforderm in termsofthefirst
genericbinaryinterval
anyrecourseto orthonormality.
in our(6.3) et seq., butwithout
of
offiniteproducts
theintroduction
Renyi'sapproachcan be usedto motivate
to deal with
of the Walsh functions,
Rademacherfunctions,
and, therefore,
in
itis also possibleto employtheseformulas
However,
strings.
involving
problems
19991
121
anotherway to establishan unexpectedconnectionbetweenthe multiplicativity
of
the Rademacherfunctionsand theirstatisticalindependence.
Recall that,in Section2, we used thestatisticalindependenceoftheRademacher
functions,
togetherwiththe vanishingof theirmeans,to justifythe Multiplicativity
Formula (2.1). However,as noted by Kaczmarz and Steinhaus,this formulacan
also be established directlyby calculus, without appealing to the notion of
statisticalindependence[4, p. 125f].
To do so, note firstthat(7.1) impliesthat
rj = r1oTT-1 for]j E N,
whereT is the binaryshiftgivenby(5.1). For fixedj, let 1 varyin the rangefrom0
to 2j-' - 1, and denote by I, the l-thdyadicintervalof orderj - 1. Since Tj-1 is
linear on each such I, and maps it onto fl, while the derivativeof T;1 on I, is
2i-1, the change-of-variable
formulagives
r o Ti"- (
fr1(2w)dow
I
) dw =
r,(w) dw = 0
(9.6)
foreach I,. Summingon 1 thenyields
f'rj() dw = 0 for]jE
N,
(9.7)
as assertedat the startof Section 2.
To arriveat the Multiplicativity
Formula(2.1), we can now employan argument
used by Kac in a different
connection[2, p. 27]. Order the subscriptsso that Ik is
the largest,and thenwrite
&J)rj2( &J)
...
f1r1(w)r1(w)
|Orjl(
rik(
d
) d
-1
2ik-1
=
E
| JSJ(t)ri2((0) ...
r1()1w
2j1E
())d,
rik(
1w dw,
where the I, are the dyadic intervalsof order Ik - 1. Since the first k - 1
functionsrjl,r1_,
... , r
are constanton each such interval,while,by (9.6) with
j = Ik,the integralof rik over each one is zero, the integralsin the sum at rightall
Formula follows.
vanish,and the Multiplicativity
With this resultin hand, we can establishthe statisticalindependenceof the
Rademacherfunctionsbyproceedingin the followingway.We integrate(9.2) over
fl and use (9.7) to findthatforeveryj E NJ
1
Pt r = zyI
(9.8)
-2whenevery,= +1.
Integrating(9.3) over fl gives
P{rt = 'y,,**, rm=
I)
=
ffl
(+2
)
Now integrating
Formula along with
(9.4) and makinguse of the Multiplicativity
(9.8) gives
m
1
=
=
HP{r
P{r=
(9.9)
yi)
rm
X
yj}.
]1
2
Since m E NJis arbitrary,
it followsthat multiplicativity
of theRademacherfunctions
impliestheirstatistical
independence.
122
[February
The precedingreasoningis not limitedto the case of Rademacherfunctions.As
a furtherapplication,recall fromSection 7 thatthe Walsh functions
...
* I** W2k ...
Wj , W2j
(9.10)
multiplicative.
When j = 1, these funcare, for fixedj E%N and varyingk E %N,
tions reduce to the Rademacher functions,as is evident from(6.1) and (6.2).
Notingthattheyhave mean zero, we can repeat our argumentand conclude that
independentforeveryfixedj E N.
the functions(9.10) are, in fact,statistically
(9.5) in (9.9), and do the same in (9.8), then
we make the substitution
If, finally,
(9.9) becomes
m
1
= n P{aj = aj}.
P{a1 = a,,..., am= am) =
Thus, statisticalindependenceof the binarydigitsis, itself,a directconsequenceof
the two notionsare
of the Rademacherfunctions,and, therefore,
multiplicativity
equivalent.
entirely
10. EXTENSION TO GENERAL BASES b. All the reasoningof the preceding
section can be carried over to integralbases b > 1, once we find a suitable
generalizationof Renyi'sidentity(9.1). To this end, let -yand 7qbe b-throots of
unityand considerthe expression
b-i
E yb-dd
=
d=O
d
b-1
E
_1
d=O
')
Since yb = _qb = 1, the quotient-q/y is also a b-throot of unity,and, therefore,
the right-hand
sum vanishesunless -y= 7q.Consequently,
b-i
1+ E
eb-d
d
d=1
b
[q
and we have foundthe identitywe need.
Proceedingas before,foreach j E NJwe get
b-i
1+
E yb-drd
d=1
b
=Xrj
(10.1)
Yj}X
and, since,by (8.3),
d1... I b - 1,
da = Oforj E- N and d
frd())
(10.1) gives
integrating
JX{rj=
}
j} = Pr=
(10.2)
=
This time,
b-i
m
X{r1=yj.. S=S
rm=yn,}
[
_ 1 r+=dj}
Xm
m
1 +
E
j-d d
B
(10.3)
Expandingthe second product,we get an expressionof the form
1
of the rj),
bm(1 + sumsofweightedpower-products
(10.4)
wherethe weightassignedto each rj is )jThus, integrating
(10.3) over fl and
Formula (8.5) for the b-adic Rademacher functions,
using the Multiplicativity
yields,in view of (10.2),
m
1
HPr = y1}
(10.5)
P{r1 = yi,...,rm = Ymi)
Accordingly,multiplicativity
of theb-adicRademacherfunctionsimpliestheirstatistical independence,
just as in the case b = 2.
In Section 8, we introducedsubsetsof b-adic Walsh functionsof the form
(10.6)
Wj, Wbi, ..., Wbkj, . . .,
for fixedvalues of j E NJand varyingk E-N, and pointed out that theywere
multiplicative,
in the sense that
Wbjlj(
(I)
...
Wbjkj(
WbJ2j( )
(I)
dw =
0
wheneverthe exponentsare distinct.We can now show that theyare statistically
independentforeach j E N byextendingthisformulain the followingway.Let b'
be the smallestpositiveintegersuch that WJb 1. Then, as a consequence of (8.5)
in the extendedsense
and the choice of b', the functions(10.6) are multiplicative
that
I
d()d
.().
w*
bk(( w)
dw = 0
foreveryk E N, where the d's are restrictedto the values 0, 1, . . ., b' - 1 and do
not all vanish. It is thus possible to apply the foregoingreasoningto them,by
replacingb by b' in the formulasabove. This yieldsthe statisticalindependenceof
the functions(10.6), and it generalizes (10.5), which,in view of (8.6) and (8.7),
correspondsto the case j = 1.
Returningnow to (10.3) and setting
=
exp(
2y3')
(10.7)
for ] 1 . . ., m, with fj3Ee {0,1, ...,b - 1}, we get, in view of (8.2) and the
expansion(10.4), a formulaforthe indicatorfunctions
= P3,n}
X{b 1= ,l 1 . . ., b,m
of the rj. The
of b-adic intervalsof order m, in termsof weightedpower-products
resultingformulais, clearly,comparableto our (8.5), et seq., and it can be used to
motivatethe introduction
of power-products
of the b-adic Rademacherfunctions,
and, thus,of the b-adic Walsh functions,in orderto deal withproblemsinvolving
b-adic strings.
It can also be employedto establishan equivalence betweenmultiplicativity
of
the b-adic Rademacher functionsand the statisticalindependenceof the b-adic
digits,generalizingthe case b = 2 treated in the previous section. Indeed, it is
124
[February
enoughto make the substitution
(10.7) in (10.5) to get
P{b=
b.
=
m
1
bt= = FI1P(bj =[3.j
1m =1
Consequently,multiplicativity
of the b-adic Rademacherfunctionsimpliesstatistical
independence
of theb-adicdigits.
in extendingto a generalbase b the argumentused
Since thereis no difficulty
in the previoussection to establishthe Multiplicativity
Formula for Rademacher
functions,withouthavingrecourseto theirstatisticalindependence,we conclude
that statisticalindependenceof the b-adic digitsis equivalentto the multiplicativity
property
(8.5) possessedbytheb-adicRademacherfunctions.
One wayto look at thisresultis as follows.Recall that,in Section 8, we derived
the Multiplicativity
Formula (8.5) fromthe formula
w)) dw =
fl exp(i1ttjbj(
H f1exp(ipjbj ( w))
dw,
(8.4)
by setting,tj = 2iTdj/b,where the dj are integersin the range from0 to b - 1
and j varies from1 to k. As shown there,(8.4) is a directconsequence of the
statisticalindependenceof the functionsbj.
However, it is known that the validityof (8.4) for all real values of the
real
parameters ,aj impliesstatisticalindependenceof the otherwisequite arbitrary
functionsbj, and Kac himselfwas among the firstto prove it, by use of Fourier
analysis[3].
In effect,what we have foundis a refinementof Kac's sufficiency
result,in a
more algebraicsetting,wherethe additivegroupof the reals has been replaced by
the group of integersmod b. For, in restricting
the bj's to take integralvalues in
the range from0 to b - 1, instead of arbitraryreal values, we have established
that the validityof (8.4) for the narrowrange of values of ,Ujindicatedabove is
alreadyenoughto ensure statisticalindependenceof the bj's.
The argumentwe have used cannot be regarded as new, even tho-ughits
applicationto Rademacherand Walsh functionsis. It is a reductionto the case of
the cyclotomicgroupof a methodsuggestedbyRenyi[9, p. 171] forprovinga more
general resultabout the statisticalindependenceof familiesof randomvariables,
attributedto Kantorovic. Renyi would, no doubt, have appreciated that the
method he proposed could be applied, trivially,to the classical Rademacher
functionsin orderto establishtheirstatisticalindependence,but he seems to have
been unaware, as was Kac, of Mendes-France's extensionof the Rademacher
functionsto the b-adic case and, thus,missedthe opportunity
to applyhis method
there.
The reader interestedin learning more about normal numbers can consult
Niven's Carus Monograph [7] and the treatiseby Kuipers and Niederreiter[5],
whichcontainsan extensivebibliography.
in partbytheConsiglioNazionaledelle Ricerche
Thisworkwas supported
ACKNOWLEDGMENT.
Research.
andtheItalianMinistry
oftheUniversities
andScientific
andTechnological
REFERENCES
e leursapplications
Rend.Circ.Mat.
1. E. Borel,Surles probabilites
denombrables
arithmetiques,
Palermo27 (1909)247-271.
2. M. Kac, StatisticalIndependencein Probability,
Analysisand NumberTheory,Carus Math. Mono-
graphs,
no. 12,Wiley,NewYork1959.
1999]
125
3. M. Kac,Surles fonctions
independentes,
I, StudiaMath.6 (1936)46-58.
4. S. Kaczmarz,
and H. Steinhaus,
Theorie
derOrthogonalreihen,
Monografie
Matematyczne,
vol. 6,
WarsawandLwow,1935.
5. L. Kuipers,andH. Niederreiter,
Uniform
Distribution
ofSequences,
PureandAppliedMath.,no.
29,Wiley,NewYork,1974.
6. M. Mendes-France,
Nombres
normaux.
Applications
auxfonctions
pseudo-aleatoires,
J. d'Anal.
7.
8.
9.
10.
11.
Math. 20 (1967) 1-56.
I. Niven,Irrational
Numbers,
CarusMath.Monographs,
no. 11,Wiley,NewYork,1956.
S. S. Pillai,On normalnumbers,
Proc.IndianAcad. Sci.,Sect.A 12 (1940)179-184.
A. Renyi,Foundations
ofProbability,
Holden-Day,
San Francisco,
1970.
D. D. Wall,OnNormalNumbers,
Ph.D. Thesis,Univ.ofCalifornia,
1949.
Berkeley,
H. Weyl,Uberdie Gleichverteilung
vonZahlenmod.Eins,Math.Ann.77 (1916)313-352.
GERALDS. GOODMANtookdegreesin mathematics
at Haverford,
BrynMawr,andStanford,
where
he was a pupilof Loewner.His doctoralthesiswas devotedto Loewner'stheoryof transformation
semigroups,
appliedto conformal
A chanceencounter
mapping.
withRenyiled himto use thoseideas
to obtainnewresults
Markovchains.Lateron,he appliedprobabilistic
concerning
toolsto Loewnerian
semigroups,
suchas totally
matrices.
he haspublished
control
positive
Although
papersinrealanalysis,
hisfinest
theory,
andfractal
imagegeneration,
someconsider
mathematical
achievement
wasfinding
a
permanent
postin a Renaissance
city.
Statistics
Department,
University
of Florence,VialeMorgagni,59, 50134 Florence,ITALY
goodman@ds.unifi.it
.. . it did not take the Miloslavskyslong to find a pretextfor
was foolishenoughto
Matveev.Thateducatedgentleman
arresting
be foundwitha book of algebrain his baggage,whichwas,
takento be a formofblackmagic.
naturally,
EvenNikita,whenhe heardofthearrestofhismentor,
couldonly
"He was askingfortrouble.Whatdid
shakehishead and remark:
he wantwithsuchstuff
anyway?"
Edward Rutherfurd,
Ruisska,TheNovelofRussia,CrownPublishers,Inc.,
New York,1991,pp. 347-348.
ContributedbyGeorge Dorner,WilliamRaineyHarperCollege,Palatine,IL
126
[February

Statistical Independence and Normal Numbers: An Aftermath to

Transcription

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