NATIONAL BUREAU OF STANDARDS REPORT

2220

THE STABILITY PROBLEM FOR A THEOREM OF CRAMER

BY N. A. SAPOGOV

Translated by Ida Rhodes

Edited by Eugene Lukacs

U. S. DEPARTMENT OF COMMERCE

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NATIONAL BUREAU OF STANDARDS REPORT

NBS PROJECT NBS REPORT

1103-10-1107 23 January 1953 2220

The Stability Problem for a Theorem of Cramer

by

N„ A. Sapogov

[Published in Izvestiya Akad. SSSR Ser. Mat. 1 5, 205-21 8(1 95l

]

Translated by

I. Rhodes

The publication, reprint

unless permission Isobti

25, D, C, Such permlss

cal ly prepared If that i

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

art, Is prohibited

irds, Washington

has been speclflI for Its own use,

The Stability Problem for a Theorem of Cramer

by N. A, Sapogov

(Presented by the Academician , S® N® Bernstein)

Abstracts If the son of two random variables and X

2

is almost normally dis t ribuled , then each of these random

variables is also approxima tely normally distributed, provided that X.| and are either independent or are dependent in

a manner specified in this paper. The degree of approximation of the distribution of the summands to the normal distribution is evaluated.

I ® Int roduc tioa

1 o If X^ and are two independently and normally distributed random variables, then their sum X = X, + X

2

is also

normally distributed® This is one of the most elementary

theorems in the theory of probability® The converse proposition, namely that X^ and X

2

are also normally distributed

if their sum X is normally distributed, is far from being

elementary® This proposition was first conjectured by

P® Levy and proved in 1936 by H, Cramer [1], In addition

to the proof given by Cramer, there is an alternate proof

due to S, N» Bernstein [2], pp® 4-27-30® Bernstein uses the

more elementary theorem of Liouviile instead of Hadamard‘s

theorem on entire functions which Cramer utilized®
' .. ,

-

.

,
2

However, neither the original proof of Cramer, nor its

variant mentioned above, allows os to arrive at any definite

conclusion regarding the type of distribution of the random

variables X^ and X 0 , if either the distribution of their sum

X is not exactly --but only approxima tely—normal , or if the

variables X^ and X^ are not entirely independent . The present paper studies these questions, A previous report on

the result of this investigation is contained in a note by

the author [3].

2. The main result of this investigation may be stated as

follows i

THEOREM? Let

— X^ + X2 (1 * 1 )

b© the sum of two independent random variables and assume

that the distribution function F(x) of X satisfies the con- dition

F(x) - 1 42tt J-00

x _ 1+2

e dt |

< e , 00 <x < 05 ( 1 , 2 )

where e < 1 is a given positive number; let also (x) be

the distribution function of X, , and let

J

x dF-^x) * a-^

- M

Then

J

x

2 dF^lx) - (J x dF

1

(x ))

2 = cr^ > 0, N « Jin ~r

(x-a, ) 2

iC - " J

F (x) - —±-__

1

a, -/2W

e V dx < Ca"' (in 7)'*

1

e

, -CD <X < 00

(1.3)

3

where C is a constant which does not depend on e , cr.j or „ An

analogous statement can be made regarding F^Cx), the distribution

function of X20

However 9 the statement ( 1 c 3 ) is not the ultimate result obtainable and is subject to improvement,. At the end of this paper,

a generalization of the above result is discussed for the case

where X^ and X^, are dependent; several other observations are

also made„

II o Redue ti on to Bounded Variables

3o We shall assume that the mediae of is zero. This is

no loss of generality for if m^O, then we may investigate X^

instead of X^ and XP + instead of X

2

<> Let now

P{X, <0)1^ < 0} > j

. (2o1)

The notation P{ } indicates the probability of the event shown

within the curly brackets

„

It is easy to show that under these conditions the median

of the value X

p

satisfies the inequality

|o2 | < 1 , (2.2)

for any nonnegative £ _< l/20. In faet ? from the definition of

a median 9 it follows that

P { X

2 < m

2 } < i , P { X2

< m

p } > \

o

Consequently ? taking into consideration also (1 ,1

)

s (1 0 2) and

(2o1), we have

\

< P { X, <0; X? < m

2 }< ? {X

1 + X

2

< ra 2 } =

= ?{X < m

2 }< 7==f

2

<3t + e , 00

from which it follows that

e “it2

dt > ^ > i~h = °”2 o

This leads to the inequality

*+

a > - 1

Moreover,

P {X1 + X

2 < ra 2 } < P {X

1 < 0 } + P {X

2 < m2 }

- P {X

1 < 0 } .P { X

2

< m

2 } < | ,

since

U + V “ UV < 9

if

0 £

u <2 and 0 < v < p

Consequently

^

> P {x < m

2 }>

^ j“V* t2 dt - e ,

from which we have

—_X_ f ^ e

8 dt < 4

•J2v -oo This leads to the inequality

-oo

±- = 0 . 8 , 20

m0 < 1

4. We shall have to produce a number a > 0, such that

(a) - F^ (-a) _> 2 and F

2

F

2 (-a) > ± .

We shall show that we may take a = 3? if e

<L 1/20. Let us

< a 2

} < \ , P{|X2 | < a 2 } > \ .

choose a 0 such that

(2.3)
• • ;

• s

•. '•

••

. v. .
5

Then

P{!x

2 i- a2 ^ — 2

•

Consequently, at least one of the following inequalities

is trues

P {^2 —

"a 2^ — b

(2.4)

P {X2

> a 2

) — Tf

• (2.5)

Let us assume the hypothesis (2.4). Taking into consideration the expressions ( 1 . 1 ), ( 1 . 2 ) and ( 2 , 1 ), we have

— a. 2. ,i_2 5

i P {X, <0; X

2

< -a 2

} < P {X < -a 2) < jmL J

e

" 5t<H+ e.

so that

i f

2 a-^dt > 1 - = 0,075.

•J2

V

r*- 00

Therefore

< U5 < 3 (2.6)

The hypothesis (2.5) is treated in a similar manner and leads

to the same result ( 2 . 6 ).

From (2.3) and (2.6) it follows that

F

2 (3) - F2 (-3) > \

.

A similar inequality is true for (x)

.

For, let us choose

a Q satisfying the condition

p <i x

1

1 < %} < b p ti x

i

I i a

0 ) - b (2 * 7)

p {|X, i > a

o

> > i , Then

6

and two cases arise

or

p {x, < - a 0 } > 1

P{X, > a 0

! > ^.

Both cases are analogous. Let us fix our attention on one

of these, say the first case. We have

g

< P {X

1 < »a

Q

; X

2

< m

2

j < P { X < -a

Q + ra 2

} <

- a2i?"2 ii.2 < —.1 . e

2 d t + £ s - J

-00

therefore

r’fUt2 e d

1 1 _

this leads to the relation

t > 5 - 25 = 0.075;

o

^ 1 <* ^

From this inequality and (2*2), we obtain

a0

< 3 -

Therefore, in view of (2.7)

F

1 (3) - F

1 (-3) > \

( 2 . 8 )

5* Let us introduce—ins tead of X^ and X

2—two new variables

X^ and X

2 , such that

X? = X. when I X. I < N, i i

1 i

1 — 7 Xf = o. when I X. j > N, i — where N = *^/fn \ .

( 1 = 1 , 2 )

Clearly X* and X

2 are also independent
.

r

-

'

j;.

mm
7

We denote by F^*(x) and F

p *(x) the distribution functions

of the variables and X0

* respec lively s and. by F'^x)*

the distribution function of the sum

X* “** — Y 4_ Y /a

^

® A

p

O We note, first, that

|F*(x) - F(x) |< [ S <SF

1 (x) + I dFp(x) ] = A. (2.9)

/X(>/V |X|>A/

This results from the fact that the probability of the inequality

X ^ X*

does not exceed A, Let us evaluate A, Since

p f x

n <0; X2

< y} < P {X < y} < t + /

e"^

2

dt

,

then oo

x 2

F 0 (-N) < 2 8 + -== T e“2t dt . s12tt

N Similarly, from

P f

X

1 > 0;X2

> y J < P { X > y} < 8 + -

we obtain

. 00

i 2

L= r e-*t dt

27T Jy

Therefore

1 - F 02

(N) < 26 + —

>/2ir j e st dt.

j d F

2 (y) < ^

+ 35=-

lyl>N

4 rw

.-it2

V/N d t . (2.10)

In the very same manner , —remembering (2.2)~~, we find

that CO 2 -ft dt

.

j dF

1 (y) <4.: + j

/y/>v a-i

The inequalities (1.2), (2.9)? (2.10) and (2.11) lead to

( 2 . 11 )
' :: '

' '

'

, ' v ;> l i.i •> 8

i,

• ’ ,

"

' S KM. : ’•

. n

. . ; • - .

1 i

' ,

»
8

the relation

1 -f? ii 1- |F*(x) - -— i @” 2 dt | /2n . 'oo < 9B +

oo 8 r -/St m_i

(2.12)

III. Investigation of Characteristic Functions

6 0 Let f*(z) be the characteristic function of the variabl<

X*

isX* ) _

00 f*(z) = ECe***"' =

j e lzx dF*(x)

- CO Since | X*

|

< 2N, then f*(z) is an entire function of the

complex argument z. Similarly, the characteristic functions

f*(z) = E(e

izXl) and f

*

(z) = E(e

iZX

2) (3.1)

are also entire functions.

Our immediate goal is to find a lower bound for the

modulus Jf*(z)J , when z is in the circle

N

l z I 1

T ~ B

" B

1

It is well known that

-is 2 _ i V2FfI

oo . X 2 1ZX - aX , e e dx

S3 so that e

2 is the characteristic function of the normal

distribution

JL

-/Sac J

x x 2 2U du = $(x). I-00 We have therefore

9

X 2 2N “2 Z |f*(z) - e~

2

j ^ J e1ZX

d|F*(x) - $>(x)j +

-2N

V2it

I e xzx e~z x

dx

lx | £ 2N

(3.2)

The first term of the right-hand side is now evaluated:

2N . f elzx d[F*(x) - $(*)]

-2N

. om 2W

|{e lzx

[F (x) - fj)(x)]j - J

[F*(x) - (f(x)]iz e ±zx dx| £

-2N ~2N

<

2e + £ e = 2hyAa + £}> (3>3)

In (3.3) it is assumed that |z| ^ T = N/8 and ^

is defined by (2.12),

The second term of the right-hand side of inequality (3.2) is

evaluated in the following manners

CD 1

yac

\X\

f izx

J e e ^2N

-±y:z

2 dx

< I

rl

4-T 2 00 -e

r

.. - u

2

/

2

5 2N- T J

2N-T

/2i

Tx - j&L , e dx <

2N

*> < h

°- n /h

( 3 .4 )

Putting together the inequalities (3.2), (3.3) and (3.U), we obtain

, i_2 2 n „2 , „„2

(z) - e“ 2 '

| < 2*/ A(l + £) e -7N‘/4

< 18 6 +

16 %(N»1) 2

\ N

2

/!^ N

; \ IT/4. IT. 1

</*F(n-i)

c (1 + T)+?n

+ N

2e-21N2/128

+ e7N2/lt =

186^ in |

+6

21/128 l„’i + 6

7A

8 S

•7N

2

/U <

(3.5)

1 0

provided N = sjln \ is sufficiently large and |z| < T = ^

In the circle under consideration the following relations

hold

-£ z 2

. -N/128 1/128

e > e =8 (3*6)

From (3.5) and (3*6) we obtain

-z

2 /2 |f*(z) - e'“

2/2

I <i\e~h2

(3.7)

Equation(3»7) is true when fzj < T 9 and if 6 is so small

that

i m i + e

21/ 128 m |

+ < * s 1/128

Therefore in the same circle |z| < T

| f*(z) | > \

e‘*

|z2

' (3.8)

From this we conclude that i*(z) has no zeros in the

circle j z | < T»

7. According to (3.1) we have

f*(z) = f.j*(z) f

2* (z) *

Therefore both functions f^*(z) and f 2*(z) have no zeros in

the circle |zj < T 9 so that their logarithms

^(z) = In f *(z) and /

2

(z) = In f

2

*(z)

are regular functions

1

1

From ( 3 * 7 ) it follows that

|f*(z)i= | f,*(z) f *(z) | < |

e * 1 >1 z| < T, (3.9)

Let z = t + is, where t and s asje real numbers. Then, in

view of ,( 2 . 8 ), we have

F*(3) - F*(-3) > \ and therefore

co

„ r -sx

f, (is) = J e dF. 1 -no 1^(x)> ) e SX

dF

1

*(x)> ie 3|S

* , •oo ~ - 3

From (3.9) and (3.10) it follows that

|f*(z)|^ f *(is) = x 2 2 % ( is ) Similarly, we find that

*/. \ _ f*(is) ^ 31 z f + ilz2

f 3e‘

(3*io)

(3*11)

I ** (z)|< 3e3 i z|+ ^l z2

| , |z| < T . (3.12)

Moreover, the inequalities (3*8), (3*11)? and (3*12) allow

us to evaluate the moduli jf*^(z)j and j f * ^

^ z ) ( • in fact,

5

e”^i z

J < |f*(z)j = jf*(z) l 2*(z)|;

therefore, and because of ( 3 * 11 )

|f *(z)| > e"3|z| - |z | 2

1

|f2*(a)| °

Similarly, by utilizing (3*12), we obtain

|f

2*(z)|>^e-3|z| - |Z|2

, |z|£T

8 . Let us note first s that

(3-13)

(3.14)

l^i (ln i)V3 (3.15)

12

We see then from the inequalities ( 3 * 11)51 (3*12), (3*13)

and Gol 1*) that for large values of T, the logarithms of

f.j*(z) and f

2

* ( 22 ) d© not differ much from certain poly4/t" noraials ,of the second degree, as long as |zf < Let us use the well-known formula representing a function

which is regular within a circle, by means of its real part

given on the circumference of the circle (Schwartz's

formula) ZV

f (z) iv(0) + 1_

2w u

o u(R ? $) IJlJl

l - Z

d0 ,

where f(z) = u(r ? ty ) + iv(r, ) is a function regular in the

circle |z| = |re^^|< R, and £= He^ ^ • Applying this to

the function 0^ (z) = In £,j*(z) and assuming R = T, we have

2..TT

(8,(1) = i Im 0,(0) +3?f j ln|f

1 *(|)| |4fd0,

O

whence ^7T

0;' (*) = I J

&0

(£- 2 ) From the inequalities (3«.12) and (3*13) it follows that

for any

I In (1, (1)! I < (III + 3) > m< t.

G. 16 )

(3.17)

Let us recall that we are considering only those values of

z which according to (3.1 5) > satisfy the condition

< 4t (in i) - 1/3

(3.18)

13

Keeping in mind (3«17)> we have from (3°l6)

\r: (z)|. < •

1 (t - \ xir

Whence, ,in view of (3*18), we have for small values of £ > 0

c.T3 c w<7 1 = r

(c^ are constants)®

After three successive integrations of this inequality,

we obtain

1^(2) " a

±

~ 1 P±

Z + 2 ^-j 2 I < , where h’*! are certain constants, jz | £

(3.19)

A similar inequality may be derived for 4^) i t ^ie

logarithm of the characteristic function f

2 *(z) of the

variable X

2

* •

IV* Proof of the Basic Theorem

9. Since ^(0) = 0, it follows from (3°19) that

l°il <

-tttherefore

Consequently

,

.2 , ,

2 ciT

i

3

4±

(z) - i 0-jZ + i^sr I < "y~~ •

*, N _

4,(z) „ >M~ i*,z2 + h(z) = e where H(z^ is regular for [z | < T.

( and in this circle

T

3

1 "f” - 2c

1

| H(z) | < 2c

since T^<>/To But in this case

where

eH(z) = 1 + H

1 v^/ 9

(2) = X 2 + X 0 z + .

.

1'“ # *1" 2

is a certain entire function, such that in the circle

(*+. 1 ) M < T,

I

' |H

1 (z)| < c 2

-Jr (**.2)

Therefore

* V>)= eip ' Z ' ir‘ Z (i*H

1 (z))=[l*(ip1

z

Here

* ...](l + XlZ+ X

2

z

2

+ ...).

x = -1- j

H1 < z ) dz

(4.3)

X = _1_ j

H±(z)d.z

1 2itiizi=T

1 z 2 2 2iti |z | =T2. z C onsequently, considering (4,2), we find

l^il <0

8 TT> l^l '"s' "< ItP

But as an entire function f 1 *(z) allows an expansion

where

f

l*

(Z) = 1 * ia

i

Z “ if z2 +

a

R = M[ (X

1 *) k

], k = 1,2,...

k

k • z +

’ • ’ *

Comparing this expansion, with (4,3)? we obtain

V&i - 2iPixi-^2 = a

2

Remembering the previously obtained values of X-^ and X^ , we have rj l“ a

ll = i xii < e 2T

and for small £ > 0

is the var2 2 2 since /T —>0, as £ 0. Here <r

-j = a 2 ~ a^

iance of * . From the above results, it follows that

, fp3

|^l(z) - i 3-jZ + z

2

j < 2Cl ^ ^ ^

. (4.4)

A similar inequality is true for ^(z) = In f 2*(z).

by

, x iat z-|^2

z

2

g 1 (z) = e

*

Then we conclude from (4.4) that

Denote

fl*(z) = g^(z)(l + H

2

( z )) , (M-.5)

where H

2

(z) is an entire function, such that in the circle

|z ' ^ T

1 T 3

|h2

(z) I < c 6 . (4.6)

Moreover, H

2

(0) = 0, since f^*(0) = 1.

10. The relation (4,5), which indicates the degree of closeness between the characteristic functions f.*(z) and g^ (z) , also allows us to evaluate the deviation of the corresponding

distributions from each other. For this purpose, we shall

make use of the Theorem of Esseen [4], See also [5] PP*

21

2

-21

4

„

THEOREM (Esseen)o Let A, L, and X be positive constants.

Let F(x) be a non-decreasing function and G(x) a function

of bounded variation. Ifs

16

1) F(- oo )= G(-oo), F ( oo ) = G(o°) ; o°

2) J |F(x) - G(x) I dx < oo • - 00

3) The derivative G* (x) exists for all x and |G'(x) j < A;

f | ill)., ait ).

| at = * , —

L

where f(t) and g(t) are the characteristic functions of F(x)

and G(x) respec lively , then

| F(x) - G(x) | < k + c(k) £ ,

for any k > 1, where c(k) is a finite positive number, defined by k.

This theorem is immediately applicable to our case, if we

assume

L = T

1

, £(t) = fj(t), g(t) = g 1 ( t)

In fact, from (4.5) , we have

f-j* (t) - gx

(t)

T l^-l i

for any real

function and

t 9 Since H

2

(0) = 0, then H

2 (z)/z is an entire

in view of (4.6), the following is true for

Since the modulus of a regular function reaches its maximum

on the boundary of the region, then for jz| < T^ , we also have

H

2

(z) z
. ...

• , -K-- If ;.

=

. n - -

.. . x \

'

•
17

Therefore

T

i f/w -g ft) x _L

-T,

f

2

i H

2 (t)

I T

l

2

a dt <J |— |*<2c6 — = X

2

Moreover, the distribution corresponding to the characteristic

>2 function g^(t), namely x -i-

[

l 7 -oo

2ol dx = G(x)

has a bounded derivative

(x“al)

<

|G,(x)l= ^ e ^

In consequence, the Theorem of Esseen allows us to state that

(x-ai ) 2

T

F *(x) L_

J

e 2al dx

1 J

X

<0 Tl

+

°8

7

~T” ^

I 9

I \ 8

00 < X <co (U.7)

In the above, we assumed the hypothesis (3*15). If that does not hold,

in other words, if ,

OL > 5(ln|) 3

,

then the inequality (k«7) is trivial, provided the constant c^ is

larger than 2^^, for under those conditions

( ln j) 8 > 1 .

In order to obtain a proof of (1*3), it is sufficient to note that

1 8

00

Fx(x) - F

1

*(x)| f dF, (x) ^ i»£ + -S-h I e

” 4"

2

du.

The often repeated condition that £ must be sufficiently

small does not have any bearing in the final formulation

of the theorem, for we can obviate this condition, by

raising the value of the constant Co

Let us note, in particular, that if > a > 0 for all

sufficiently small c, > 0, where o is a constant, then we

have

for any <x> < 1/8, provided 8 > 0 is sufficiently small,

5

o

Supplementary Notes ,

1 1 „ Let us turn our attention to the fact that the theorem

just proven may be paraphrased in terms of moments. In

fact, the closeness of the distribution function F(x) of

the quantity X to the normal distribution function indicates

that several first moments of F(x) and the corresponding

moments of the normal distribution are also close. It

follows then that several first moments of the distribution

F-|(x} and the corresponding first moments of some normal

distribution $^(x) are also elos^*

In our investigation we found that the hypothesis assuming complete independence of and was not essential.

oo

19

Let now X be the sum of two dependent variables and

X

2

, whose distribution functions are a priori (x) and F

2 (x)®

Assume that the distribution function F(x) satisfies the

condition

| F(x) - J (x) I < -OO < X < CO

,

and that the dependence between and X

2

is such that

oo

| F {X^ + X

2 < x } - J

F

i (x-y)dF 2

(y) | < £ 5 -oo < x < oo . ”00

We consider two independent variables X,

( and X

2

with distribution functions (x) and F

2 (x) , and assume that

X.| + x

2 = X 5 We then have

|F(x) - | (x)

|

< 6*+ 6 "

,

where F(x) is the distribution function of X® Now we may

apply the Theorem of Section II® A corollary of Cramer's theorem:

Let the sum X of two random variables and X2 be subject to the Gaussian law but the two addends may, in general, be dependent® If there exists a constant ja, such

that x.j and x

2 “aX^ are independent , then and X2 are

normally correlated®

Indeed, in this case the quantities ( 1 +a)X^ and X

2~aX^

are independent, and moreover, their sum is equal to X

and is normally distributed® Then each of the above quantities is individually subject to the Gaussian law, and it

20

follows that and X

p

are connected by a normal correlation,,

Finally, it might be ©f vale© t© note a fact which,

though it is not in the direct line of our investigation,

is closely connected with Cramer 9 ® theorems Similar to the

Gaussian distribution, Poisson”® distribution possesses a property, stated in a theorem of D« A» Raikov [61#. which is

completely analogous to that of framer* At the same time,

both of these distributions are the limits ©f binomial distributionSo For these, a theorem holds which is analogous

to Cramer's and flaikov's® That is§ if the sum of two independent random variables is binomially distributed, then

each addend is also binomially distributed (or is non- random) «, This results from the fact that the generating function

(p+tq)° of the binomial distribution has polynomial divisor's of the same type e

Received April 24-, 1950

21

Biblioq raphy

1 . Cramer, N. liber eine Eigenschaft der norraalen Verteilungs

function. Math, Zeitschrift, 41, 405-414, (1936)

2. Bernstein, S. N, Theory of Probabilities, Moscow-Leningrad, 1946,

3. Sapogov, N. A. On a property of the Gaussian distribution, Doklady Akad. Wauk SSSR (N.S) 73? 4-61 —^+62

,

(1950),

4. Es seen C, G, Fourier analysis of distribution functions.

A math etna tical study of the Lapiace-Gaus sian

law. Acta Math., 77? 1-125 (19^5).

5. Gnedenko B. V. and Kolmogorov A, N. Limit distributions

for sums of independent random variables.

Gosudarstv. Isdat, Tehn. Moscow-Leningrad 1949

(204 pp)

.

6. Baikov, D. A. Or? the decomposition of Gauss and Poisson

laws, Xzv. Akad. Nauk, Leningrad SSSR, Ser.

Mat. (2) 91-124 (1933).

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