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NATIONAL BUREAU OF STANDARDS REPORT

25^5

A PROPERTY OF THE NORMAL DISTRIBUTION RELATED TO

A THEOREM OF S. BERNSTEIN

by

Eugene Lukacs and Edgar P, King

<NBS>

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NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-10-1107 1). June 1953 2545

A PROPERTY OF THE NORMAL DISTRIBUTION RELATED TO

A THEOREM OF S. BERNSTEIN

by

Eugene Lukacs and Edgar P. King

PREPRINT

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unless permission Is obtain 25.D.C. Such permlssior

eally prepared If that age

Approved for public release by the

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Standards and Technology (NIST) on October 9, 2015

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s, Washington

s been specifiir Its own use.

A PROPERTY OF THE NORMAL DISTRIBUTION RELATED TO

A THEOREM OF S. BERNSTEIN

by

Eugene Lukacs and Edgar P. King"

1. Summary . The following theorem is proved.

Let X^ , X2 , Xnbe n independently (but not necessarily

identically) distributed random variables , and assume that the

n^b moment of each X^ (i = l,2,...,n) exists. The necessary and

sufficient conditions for the existence of two statistically

n n

independent linear forms Yi = 2 a X 0 and Y 0 = 2 b^Xo are: 1

3=1 3 3 2 S=1

3 3

(A) Each random variable which has a nonzero coefficient

in both forms is normally distributed.

(B) 2 a sb

s of = 0 . s=l

2

Here o

s denotes the variance of Xs (s = l,2,... Jl n).

For n=2 and ap=b]_=a2 =l , b

2=-l this reduces to a theorem

of S. Bernstein [1] (see also [3]) which was also proved by

M. Kac [[(.] in measure theoretic terms. Another particular case

of the theorem is stated without proof in a recent paper by

Yu. V. Linnik [5] .

2. Introduction . We consider two linear forms

(1) Y]_ = 2 as Xs ; Y2=2 b

s X

s =1 s =1

in the n independently distributed random variables X-^ ,X2 ,

* ’

* >Xn

«

National Bureau of Standards, Washington 25, D. C.
.

.

' i

: ;

' ...

'

I

.. i 3

-

'

i V-
We arrange the variables so that the first p (X-^ ,

* * 0 ,X ) have

nonzero coefficients in both forms and the remaining (n-p)

have zero coefficients in one form or the other. Clearly

0 ^ p < n. When p=0 , Y-j_ and Y£ are trivially independent; when

p=l , Y]_ and Y2 cannot be independent. For p ^ 2, it is clear

that the statistical independence of the original linear forms

(1) is completely equivalent to the independence of the forms

P P

Zl = 2 a

s Xs and ^2 = 2 b s X

g

. This means that when p < n

s=l s=l

the distributions of the random variables Xp+]_ , •••• ,Xndo not

affect the independence of Y]_ and Y2 . This is why the theorem

contains only a statement about the distributions of those random

variables with nonzero* coefficients in both forms.

If for some pairs of corresponding coefficients, say the

first r (l<r<p), the relation

(2)

holds, then we

fl = f2 _ ... = fr

bi b2 br

can rewrite Z]_ and Z2 as

C

Zx = C(bxXx + - ••+b rxr ) + ar+1X

r+1 + "*+a

p

X

p

, and

Z2 = b

1X

1 + ---+brXr + br+iXI

, +1 + '"+b

p

X

p

Introducing the new variable X-j_ = b]_X]_ + * • •+br».Xr , we see that the

independence of Y-j_ and Y2 is equivalent to the independence of

the forms Z1 = CXL +a

p +1Xr+1

+* • • +a

p

Xp and Z 2 = xl +b

r +iXr+l +

° ° ° +b

If the theorem holds for the forms Z^ and Z2> Cramer's theorem [2]

shows that the normality of X]_ implies the normality of the

random variables X]_ ,X2» • • • ,Xr . We proceed in the same manner if

there are several groups of random variables for which a relation

- 3 -

of type (2) holds. Hence out problem reduces to the study of

the independence of two linear forms whose coefficient matrix

contains no vanishing minor or order 2.

Finally it is clear that the independence of Yq_ and is

n

equivalent to the independence of the forms Y]_ = Z a (XS -E[XS ]) ~ n s=l

and Y0 = 2 b (X -E [X- ] ) . Therefore we shall assume without d s=l S S

loss of generality that the following conditions are satisfied:

(i) as b s t 0 (s = 1,2,. .

. ,n)

(ii) a

g

b

t

- £ 0 for all s =(=t (s,t = 1,2,.. ,,n)

(iii) E [Xs ] = 0 (s = 1,2, ...,n)

•

3 • The functional equation for the characteristic functions . Denote the distribution function of the random variable Xs (s=l,...,n)

by Fs (x) and the corresponding characteristic function by f s (t).

Let h(u,v) be the c.f. of the joint distribution of Y]_ and Y2 and

write h]Ju) = h(u,0), h^v) = h(0,v). Clearly h-^(u) and h-2 (v)

are the c.f.'s of the distributions of Y-^ and Y2 , respectively.

We prove first that our conditions are necessary; that is,

we assume that Y^ and Y2 are statistically independent. In terms

of characteristic functions this means

( 3 ) h(u,v) = h]_(u)h2

(v)

Further, because X-j_

,

(k) hX

( u) = "TT f s( a s u ) »

• • • ,Xn

# • • are independent, we have

n

S=1

n

( 5 ) ^2 ( v > = TT f s( b

s v)

S=1

n

h(u,v) = IT" f s( a

s u+b

s v ) s=l

( 6 )

- 4

-

Finally, substituting ( 4 ), (5>) » and (6) , in (3) we obtain the

following functional equation in the characteristic functions:

(7) TT f s ( a s u+b s v) = "TT f

s

( a

s u ) f

s( b

s v ) • s=l s=l

4. The differential equations for the cumulant generating functions . The general procedure for determining the explicit form of the

characteristic functions f g (t) will be to differentiate the

logarithm of (7) r times (r = l,2,...,n) with respect to u,

set u=0 , and solve the resulting n differential equations for

In f s (t) (s = 1,. .. ,n)

.

We first note that f s (o) =1 ( s = l,...,n) and that f s (t)

is a continuous function of t. Therefore there exists a neighborhood of the origin in which all the factors occuring in (7) are

different from zero. This neighborhood could of course be the

entire plane. In the following derivation we restrict the values

of u and v to this neighborhood; then we may take the logarithm

of both sides of (7) and obtain

n n n

(9) Z ^s (a

g

u+b s v) = 2 ^s (a s u) + 2 ^s (b

g

v) , s=l s=l s =1

where /s (x) = in f^(x)

Differentiating (9) r times with respect

n

( 10 ) 2

s=l

/ (suu+b-v)

u=0

to u and setting u=0 yields

n

= 2

s=l dur

( a

3

U ) J u=0

Letting Z 3 = a

g

u, we find that the typical term on the left side

of (10) becomes
"

' '

,
- 5

-

( 11 ) 4(as u+bs v) = a. u=o z s =0

Employing the substitution lfs

( v) = /(b v), (11) becomes

( 12 ) r ^s( a

s u+b s v ) = ftl\ il f ( v)

u=0 y°s) dvr 3

Similarly the typical term on the right side of (10) becomes

(13)

,(s)

dur ^s

( a

s U)

u=0 z „=0

( ias

)

.1 s where 1 is the r tb order cumulant of Xs

• Substituting (12)

and ( 13 ) in (10) we obtain

Z -^r lJ s (v) = 2 (ias )V S>

dU)

S=1 dv 3=1

(r 1,2,... ,n)

where Is = e|

Differentiating (14) (n-r) times yields the system of differential equations

(15)

n

Z

S=1 Jsf

dn

dvn

n

Z t

n dn

s=l -^s dvn

(r = 1,2,... ,n-l)

n

We have to determine all the distribution functions whose characteristic functions satisfy this system of differential equations

and the initial conditions

,r (15a) d

dv2 v=0 = ub j

r/

s)

' s r

r = 1,2,... ,n-l

ts (0) = 1

V

We now define

- 6 -

Dn=

'*1

If

ll

n

S

2

• • • • • Cj n

th and denote byJ D„ the cofactor of the element in the s column

s ,n

and the n^*1 row of D n

. Considering (If?) as a system of n linear

,n

equations in the quantities —S.._ ljj (v), we obtain the solutions

dvn

( 16 ) dR

Hr dvn

D.

(v) = 2 (ia 0 ) n

/fi

s) = i n

C say

; n s=l “ s ’ n

Integrating (16) n times and employing the initial conditions

(l£a) yields

n-1 (ibq)^* / q \ _• 6 S n ~ = ~TT~ vJ + ~?r (iv) •

Since f (b g v) = exp[^ (b g v)] = exp[l|js (v)] we have

(17) f q

(

b a v ) = exp

(s) n-1 K

2 -JLj=l j

(ibv)‘ + -3

^ < ib s v ) n

b s nl

In case any of the functions f (t) become zero for some

s real t, this solution is valid only in a certain neighborhood

of the origin. We next show by an indirect proof that none of

the functions f s (t) (s = l,...,n) has a real zero; from this

we can conclude that ( 17 ) is valid for all real t.

Let us therefore assume that one or more of the c.f.’s

f s(t') have zeros. In this case at least one of the functions

f s (b v) will have a zero. Denote by v£ the zero closest to the
. iv

-
- 7

-

origin and by fr,(t) a function for which f^b^v^) = 0. For

| v | < |v°| we have f s (b s v) £ 0 (s = l,...,n) and formula (17)

is valid. Let v be a real number such that |v| < |v°[; then we

have by ( 17 )

(18) fp

(brv) = exp

n_1

i Cr n n

Z -A— (ib v) J + (ib v)

j=l 3 1 bjnl r

But f (t) is a continuous function. Hence 1^ o f.„(b v) =

v; = f

r

(t>rv°) = 0 by assumption. However, from (18) it is clear that

v 11^v

o f

r

(b rv) = exp

n-1 *(n)

2 (ib A°) J + ^

(ib v°)

r r bnn «. r r*

n

3=1 3

which is always different from zero. This is a contradiction, and

hence formula (17) is valid for all values of v. Writing t = b e v

s we finally obtain

(19) f (t) = exp

n-1 *( s ) 2 _J

3=1 3

(it) 's ,n

b

s nl

(it) n

5. Proof of the theorem . We have determined all the solutions

of the system (15) satisfying the initial conditions (15a) . In

order to find the distribution functions whose characteristic

functions satisfy this system we must select those functions ( 19 ) which are characteristic functions. This is easily done by means

of the following result due to Marc inkiewicz [6],

Theorem of Marclnkiewicz . a 0

+a-i z + . . .+ar

z

r No function of the form e (r>2) can be

a characteristic function . Hence the degree of the polynomial in (19) cannot exceed 2. In
'/

”J& O'.E

>

. , .

' - : . : •

r : :
case n>2 we must have

- 8 -

A.

(s) = 0

C s,n

= 0

Do n Because the factor -zs—

D,n

reduce to

j = 3,4, • • • ,n-l; s = 1 ,2 , . .

. ,n (n >3)

(n>2) ; s = 1,2, . • . ,n

is C cannot vanish, these relations

s ,n

(20) Aj

S> =0 j = n>3

2 ag/f^

S

^ = 0 n>2 . s=l 3 n

There is no restriction if n = 2. In view of (iii) = 0 also,

and ( 19 ) becomes

( 21 ) f

s (t) exp

2

o

s for n > 2 .

This shows that each X (s = l,...,n) must be normally distris buted, which is condition (A) of the theorem. All cumulants of

order r>2 vanish for a normal distribution, hence equations (20)

impose no additional restrictions. In case n = 2 we have

for n = 2 , where k is determined from (16) and (19). The independence of

Y-^ and implies that they are uncorrelated which yields condition

(b) and completes the first part of the proof.

It is easy to establish that conditions (A) and (B) are also

sufficient. Assuming that (A) and (B) hold, it follows that

Y]_ and Y£ are uncorrelated and normally distributed. Hence Y-^

and Y2 must be independent.

( 22 ) f

s (t) = exp

-k/2 t 2
:

1

.

\

'

; >

'

. %. ' 'i

V : ;

,9

/ ' ,

5 • -

,

•

: .. .. ..

. -i

' ?'

-

‘f i
- 9

-

For n = 2 and a

i = 8. ^ = = 1, b£ = -1 we obtain from (22)

,2 f„(t) = exp

o\ + o|

s = 1 ,2

This shows that and establishes Bernstein’s theorem.

REFERENCES

[1] S. Bernstein: Sur une Propriete Caracteristique de la Loi

de Gauss. Transactions of the Leningrad Polytechnic

Institute (I 9I4.I) .

/ " [2] H. Cramer: Uber Eine Eigenschaft Der Normalen Verteilungsfunktion. Math. Ztscht . }±1, 4.05-4U4- (1936).

[3] M. Fre'chet: Generalisations de la Loi de Probabilite de

Laplace, Annales de L’Institut Henri Poincare , 13 , 1-29 (1951 ).

[1+] M. Kac: A Characterization of the Normal Distribution.

American Journal of Mathematics . 61, 726-728 (1939)

[5] Yu. V. Linnik: Remarks Concerning the Classical Derivation

o f Maxwell’s L aw , Doklady Ak» Nauk SBSR , 8£,

1251-1254 (1952).

[6] J. Marcinkiewicz ; Sur une Propriete de la Loi de Gauss,

Math. Ztschr . Ijlt- » 612-618 (193^) •
Vf ,v |i'

.
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}))3