Foundations of the theory of probability

§

3.TheLawofLargeNumbers 61

wherethex
lf

x

2 ,

...

,

x

n

areuncorrelatedinpairs,wecaneasily

computethat

o

2

(s)=o

2

(*,)

+

o*(x

2

)+

• ••

+

o

2

(*»)

. (?)

Inparticular,equation(7)holdsfortheindependentvariablesx

k

.

§

3. TheLawofLargeNumbers

Randomvariablessofasequence

§lj

&2,

• ••
  ,

O

n,

...

arecalledstable,ifthereexistsanumericalsequence

(Zi,ct

2 ,

..

.

,ct

n

>.••

suchthatforanypositivee

P{\s

n

-d

n

\^e}

convergestozeroasn

—*

oo.IfallE(s

n)

existandifwemayset

d

n

=E(s„),

thenthestabilityisnormal.

Ifall

s

n

are

uniformly

bounded,thenfrom

P{\s

n

-d

n

\^e}-+0

»++oo

(1)

weobtaintherelation

|E(s„)-d

n\

-> «->+oo

and therefore

P{|s

n

-E(s

ri)|^

£}->0.

«->+oo (2)

Thestabilityofaboundedstablesequenceisthusnecessarily

normal.

Let

E(s

n

~E(s

n

))^

=

aHs

n

)

=

^.

AccordingtotheTchebychefF inequality,

P{|s

n

-E(

S

„)|^

£

}^^.

Therefore,theMarkov Condition

<4->0 n^+oo

(3)

issufficientfornormal stability.