Foundations of the theory of probability

§

3.The

Law

of

LargeNumbers 63

We

set

Xnl;=%k

if

I

^fc-mfc

|

^

n,

Xnk

~~

otherwise,

c*

_

Xn

+*„*+

•••

+*„

«

jrelations

k=n

ZP{|**-

*=1

m

k\

k=n

I=n

2

p

(*•»

+

**)

-*

=i

+

oo

(6)

oM^)=J>

2

(*n*)

=<>(*

2

)

(7)

arenecessaryandsufficientforthestabilityof

variabless

n

3

.

Wemayhereassumetheconstantsd

n

tobeequaltotheE(s„*)

so

thatinthecasewhere

E(s*)-E(s

n

)-»0 w->+cx)

(andonlyinthiscase) thestabilityisnormal.

AfurthergeneralizationofTchebycheff'stheoremisobtained

ifweassumethatthes

n

dependinsomewayupontheresultsof

anyntrials,

«i,%,

• ••
  ,%n

.

sothataftereach

definite

outcome

ofall

thesen

trials

s

n

assumes

adefinitevalue.Thegeneralideaofallthesetheorems knownas

thelaw

of

largenumbers,consistsinthefact thatifthedepend-

enceofvariabless

n

uponeachseparatetrial

%k

(k

=

1,2,

..

.

,n)

isverysmallforalargen,thenthevariabless

n

arestable.Ifwe

regard

$ik

=

E[EH

1

a

t

...9u(Sn)

—E

«,9l«...«*-i(

S

n)]

2

asareasonable

measureofthe dependence

of

variabless

n

upon

thetrial

E

fc

,thentheabove-mentionedgeneralideaofthelawof

largenumbers

canbemadeconcretebythefollowingconsidera-

tions

4

.

*n*

=E«,

«,...21*(

s

n)

~

E^

9t

2

...

2U-

_

i

(s«)•

3

Cf. A.

KoLMOGOROy. tlberdieSummen

durchdenZufall bestimmter

unabhangiger

Grossen,Math.Ann.v.

99,1928,

pp.

309-319(corrections

and

notestothisstudy,v.

102,

1929

pp.484-488,TheoremVIIIanda

supplement

onp. 318).

4

Cf.A.

KolmogoroY- Surlaloidesgrandesnombres.Rend.

Accad.Lincei

v.

9,

1929

pp.470-474.