Foundations of the theory of probability

§

5.EquivalentRandomVariables;VariousKindsof

Convergence

33

Asthelimitofthesequence

(x^,

4

Wl)

,•.•,

#i

Wi)

),

i

=

1,2,3,

..

.

,the

point(x

lt

x

2 ,

..

.

,

£fc)

belongstothesetU

k

.Therefore,

£

belongsto

foranykandthereforetotheproduct

k

*

§5.

EquivalentRandom

Variables;VariousKindsofConvergence

Startingwiththisparagraph,wedealexclusivelywithBorel

fields

of

probability. Aswehavealreadyexplainedin

§

2 ofthe

secondchapter,thisdoesnotconstituteanyessentialrestriction

onourinvestigations.

Tworandom variables xand

y

arecalled equivalent, ifthe

probabilityoftherelationx^=-yis

equaltozero.Itisobviousthat

twoequivalent

randomvariableshavethesameprobabilityfunc-

tion:

pu)(A) =

?(y)(A).

Therefore, the distribution functions F^ and F-W are also

identical.

Inmanyproblemsinthetheoryofprobabilitywemay

substitute for any random variable any equivalent variable.

Nowlet

X\,X%,...

,

X

n

,

... \L)

beasequenceofrandomvariables.Letusstudythe setAofall

elementaryevents

£

forwhich

thesequence (1) converges.

Ifwe

denotebyA

(

™J

thesetsof£forwhichallthefollowinginequalities

hold

K+*-*»|

<^

k=

\,2,...,p

thenweobtainatonce

A=$<§3Mj;.

(2)

mn

p

Accordingto

§3,theset

A^

alwaysbelongstothefieldgf.

The

relation

(2)

shows

that

A,

too,belongsto5-Wemay,therefore,

speak

oftheprobability

of

convergenceofasequenceofrandom

variables,foritalwayshasaperfectlydefinitemeaning.

Now lettheprobability P(A) ofthe convergence setA be

equal tounity. Wemaythenstate that thesequence

(1)

con-

verges withtheprobabilityonetoarandomvariable x,where