Foundations of the theory of probability

§4.

SomeCriteriaforConvergence 43

E

(/(*))==//(*)

P{dE)

=jf(x)

P(dE)

+//(*)

P(dE)

^f{a)P(\x\<

a)

+

KP()x\

>

a)

£/(«)+

KP(|*|>a)

andtherefore

P(l^l^a)^

E{/(

^-

/(

^

. (7)

Ifinsteadof

f(x)

therandomvariablexitselfisbounded,

1*1

^M

,

then
/(#)
g

f(M),

andinsteadof

(7),

wehavetheformula

P(|*|a«U

E(/

y

(a)

. (8)

Inthecase

/(#)

=

a;

2

,

wehavefrom

(8)

§

4. SomeCriteriaforConvergence

Let

Xi,

%2y

• ••
  yXni
  
  
  • ••
    \
  
  
  
  

*

/

beasequenceofrandomvariablesand

f(x)

beanon-negative,

even, and for positive x a monotonically increasing function

5

.

Thenthefollowingtheoremsaretrue

:

I.

Inorderthat

thesequence(

1

)

converge

in

probability

the

followingconditionissufficient:Foreache> thereexistsann

suchthatforevery

p

>0,thefollowinginequalityholds

:

E

{f(x

n+p

• *„)}<

e.

(2)

II. Inorderthatthesequence

(1)

convergeinprobabilityto

therandomvariable

x,

thefollowingconditionissufficient

:

HmE{/(*

n

-%)}

=

0.

(3)

n-*

+oo

III.

If

f(x)

is boundedandcontinuous

and/(0) =0, then

conditionsIandIIarealsonecessary.

IV. If

f(x)

is continuous,

/(0)

=

0,and thetotality of all

x

u

x

2

,

.

.

.

,

x

m

...,x

is

bounded,thenconditionsIandIIarealso

necessary.

5

Therefore

f(x)> ifx=f=0.