Foundations of the theory of probability

§

2.IndependentRandomVariables

59

thespaceR

n

,then

condition

(1) is

also

sufficientforthemutual

independence

of

the

variables

x

lf

x

2 ,

..

.

,x

n

.

Proof.Let

%'

=(x^,

x

it

,

....,x

ik

)

and

x"=

(x

h

,

x

h

,

.

..,

x

jm

)

be

twonon-intersectingsubsystemsofthevariablesx

lt

x

2 ,

...

,

x„.

Wemustshow,onthebasisofformula

( 1 )

,thatforeverytwo

BorelsetsA'andA"ofR

k

(orR

m

)

thefollowingequationholds

:

P

(*'

GA',x"c

A")=

P

(*'

c

A')P(*"c.4").

(2)

Thisfollowsatonce from

(1)

forthesets oftheform

A'

=

{x

(l

<

a

lf

x

it

<a

2

,

..

.,

x

ik

<a

k

}

,

A"=

K

<

b

lt

x

h

<b

2

,

..

.,

Af

;m

<

b

m}

.

Itcanbeshown

thatthispropertyofthesetsA'andA"ispre-

served under formation of sums and differences, from which

equation (2) followsforall Borelsets.

Nowletx

—

{x^} beanarbitrary (ingeneralinfinite) aggre-

gateofrandomvariables.

//

the

field

$

(;r)

coincideswiththe

field

B$

M

(Misthesetofall

n)

,

theaggregate

of

equations

JVi,,..../*(*i»*i.

.-•»««)=F

/Al

{a

1

)F

fli

(a

2

)...F^

n

(a

n) (3)

is necessaryand

sufficient for

the mutualindependence

of

the

variablesx

u

.

Thenecessityofthisconditionfollowsatoncefromformula

( 1

).Weshallnowprovethatitisalsosufficient.LetM'andM"

betwonon-intersectingsubsetsofthesetMofallindices

^

and

letA' (orA") beasetofB%

M

definedbyarelationamongthe'x^

withindices/xfromM'

(orM").Wemust

show

that

we

thenhave

P(A'A")=P(^

,

)P(^

,/

)

• (4)

If A' andA" arecylinder sets thenwearedealing with rela-

tionsamongafinitesetofvariables*
u

,

equation

(4)

represents

inthatcaseasimpleconsequenceofpreviousresults (Formula

(2)).

Andsincerelation

(4)

holdsforsumsanddifferencesof

setsA' (or

A") also, wehaveproved (4) forall setsof

B%

M

as well.

Nowforevery

n

ofasetMlettherebegivenaprioriadistri-

butionfunction F^(a)

;

inthatcasewecanconstruct afield

of

probability suchthat certainrandomvariables

x^

in that

field

(p

assumingallvaluesinM) willbemutuallyindependent,where

XpWillhave

for

itsdistributionfunctiontheF^(a)givenapriori.