Foundations of the theory of probability

34

III. Random

Variables

therandomvariablexisuniquelydennedexceptforequivalence.

Todeterminesucharandomvariablewe

set

limx

n

n oo

onA,andx— outsideofA.

We

have

toshowthatxisarandom

variable,inotherwords,

that

thesetA

(a) oftheelements£ for

whichx<a,belongsto5-But

A(a)=A<S<£>{x

n+p

<a}

incasea

^

0,and

A(a)=

,4©${*

n+p

<tf}

+

^"

n p

inthe

oppositecase,

fromwhich

our

statementfollows

at

once.

Ifthe probabilityof convergenceof thesequence (1) to x

equalsone,thenwesaythatthesequence

(1)

convergesalmost

surelytox.However,forthetheoryofprobability,anothercon-

ceptionofconvergence is possiblymoreimportant.

Definition.Thesequence

x

u

x

2

,

..

.

,x

n

,

..

'.'.

ofrandomvari-

ables convergesin

probability (converge

en

probability)

to

the

randomvariablex,ifforany£>0,theprobability

tendstowardzeroasn

—

oo

5

.

I.

If

thesequence

(1)

convergesinprobabilitytoxandalso

tox',

then

x

andx'areequivalent.Infact

sincethelastprobabilitiesareassmallaswepleaseforasuffici-

entlylargenitfollowsthat

p

|i*-*'i>y=°

andwe

obtainat

once that

P{x±X

'}^]?P{\x-X

'\>l

t

}

=0.

m

II.

//thesequence (1) almostsurely

convergestox,thenit

5

ThisconceptisduetoBernoulli; its completelygeneral treatmentwas

introducedbyE.E.Slutsky(see
[1]).