Foundations of the theory of probability

66 VI. Independence;TheLawofLargeNumbers

(Chap. IV,

§2)

stillhold;ingeneral,however,theexistenceof

E*|x
|

doesnotfollowfromtheexistenceofE*(#).

Toprovethatthenewconceptof mathematicalexpectation

is reallymoregeneralthan thepreviousone, itis sufficientto

givethefollowing example. Settheprobabilitydensity
f

(x)

(a)

equalto

Q

f{X){a)=

(|«|

+

2)«ln(|ii|

+2)

'

where
the

constant

C

isdeterminedby

+00

ff&{a)da

=i.

Itiseasytocomputethatinthiscasecondition

(4)

isfulfilled.

Formula

(2)

givesthevalue

E*(x)

=

0,

buttheintegral

+00

+00

j\a\dFW(a)=f\a\fW(a)da

—00

—00

diverges.

§

5. StrongLaw of LargeNumbers; Convergence of

Series

Therandom

variables

,9,,

ofthesequence

Sit*>2>•••

i

Snt••

• 
  

arestronglystableifthereexistsasequenceof

numbers

0^1

>

C^2>

• ••

  
  

  
  

("nt'•

• 
  

suchthattherandomvariables

S

n

~

Q/fi

almostcertainlytendtozeroas n

->

+00. From

strongstability

follows, obviously, ordinary stability. Ifwe canchoose

d

n

=E(s„)

,

thenthestrong

stabilityisnormal.

IntheTchebycheffcase,

c

—

*»+

*

2

J b

Xn