Foundations of the theory of probability

§

5.Equivalent

RandomVariables;VariousKindsofConvergence 35

alsoconvergestox

inprobability.LetAbetheconvergenceset

ofthesequence

(1)

;

then

1

=P(A)^limP{\x

n+p

-x\<e,p

=

0,i,2,...}^limP{\x

n

-x\<e},

from

which

the

convergence inprobabilityfollows.

III. Fortheconvergenceinprobability

of

thesequence

(1)

thefollowing

condition

is

bothnecessaryand

sufficient:

For

any

£

> thereexistsann suchthat,forevery

p

>0,thefollowing

inequalityholds:

P

{|*n+p-*n|>£}<£

.

LetF

x

(a),F

s

(a),.. .,F

n

(a),.. .,F(a) bethedistribution

functionsoftherandomvariablesx
lt

%2,...,£«,...-,x.Ifthe

sequencex

n

convergesinprobabilitytox,thedistributionfunc-

tionF(a) isuniquelydeterminedbyknowledgeofthefunctions

F

n

(a).Wehave,infact,

THEOREM

://

thesequence

x

lt

x

2 ,

...
,x
n

,

... convergesin

probabilityto
x,

thecorrespondingsequence

of

distribution

func-

tions

F

n

(a) convergesateachpointof

continuity

ofF(a) to

the

distributionfunctionF(a)
of

x.

ThatF(a)isreallydeterminedbytheF

n

(a) followsfromthe

factthatF(a)

,

beingamonotonefunction,continuousontheleft,

isuniquelydeterminedbyitsvaluesatthepointsofcontinuity

6

.To

provethetheoremweassumethatF iscontinuousatthepoint

a. Let
a'<a;

thenin

casex<

a',

x

n

==^a

it is necessarythat

\

x

n

-x

\

>a

• 
  

a'.

Therefore

lim

P

(x

<

a,x

n

^

a)

=

,

F(a')=P{x<a')^P{x

n

<a)

+

P(x<a\x

n

^a)=F

n

(a)

+

P{x<a',x

n

^a),

F

(a')^liminfF

n

(a)

+

limP(x

<

a,x

n

^

a)

,

F(a')^\immiF

n

(a).

(3)

In

an

analogous manner,we

can

prove

that

from

a">

athere

followstherelation

F(a") ^limsupF

c

(a).

(4)

8

Infact,ithasatmostonlyacountablesetofdiscontinuities(seeLebesgue,

LegonssurVintegration,1928,

p.

50.Therefore,thepointsofcontinuityare

everywheredense,andthevalueofthefunctionF(a)atapointofdiscon-

tinuityisdeterminedasthelimitofitsvaluesatthepointsofcontinuity

onitsleft.