Foundations of the theory of probability

62

VI. Independence;TheLawofLargeNumbers

Ifs

n

-E(s

n

) are

uniformly

bounded:

\s

n

-E(s

n)

\^M,

thenfromtheinequality
(9)

in

§

3, Chap.IV,

P{|s„-E(Sn

)|^}fe^-\

Therefore,inthiscasetheMarkovcondition

(3)

isalsonecessary

forthestabilityofthes

n

.

If

_

x

x

+

x

2

H

j-x

n

Sn

~

n

andthevariablesx
n

areuncorrelatedinpairs,wehave

<*

=

i*{<y

2

(xi)

+

*

2

(*

2 )+

•••

+

**(*»)}•

Therefore, inthiscase,thefollowing conditionis sufficient

for

thenormalstabilityofthearithmeticalmeans s

n

:

°l

=

o*

(

Xl)

+tf(x

2 )

+

• 
  
  
  
  • 
    
  
  
  
  

.

+

a*

(*J

=

(»*) (4)

(Theorem

of

Tchebycheff). Inparticular, condition

(4)

isful-

filledifallvariablesx„areuniformlybounded.

Thistheoremcanbegeneralizedforthecaseofweaklycor-

relatedvariables

x

n

.

Ifwe

assume

thatthecoefficientofcorrela-

tionr

mn

a

ofx

m

andx„

satisfies

theinequality

r

mn

^c(\n-m\)

andthat

c.=

2>(*).

jfc=

then

a

sufficientconditionfornormalstabilityofthearithmetic

meanssis

2

C„oi-o(HP).

(5)

Inthecaseofindependentsummandsx

n

wecanstateaneces-

saryand sufficientconditionfor the stabilityofthearithmetic

meanss

n

.Foreveryx

n

thereexistsaconstantm

n

(themedianof

x

n

)

whichsatisfiesthefollowingconditions:

P(*n<**n)

^i>

1

Itisobviousthatr

mn

=

1

always.

2

Cf.A.Khintchine,SwrZaloiforkdes

grandesnombres.C.R.del'acad.

sci.Parisv.186,1928,p.285.