Foundations of the theory of probability

§

5.StrongLawofLargeNumbers;ConvergenceofSeries 67

where the variables x

n

are mutuallyindependent. A sufficient

8

conditionforthenormalstrongstabilityofthearithmeticmeans

s

n

istheconvergenceoftheseries

2*P

(i)

n=l

Thisconditionisthebest

in

thesensethatforanyseriesofcon-

stantsb

n

suchthat

^

n=l

wecanbuildaseriesofmutuallyindependentrandomvariables

x

n

suchthat

andthe correspondingarithmeticmeanss

n

will

notbestrongly

stable.

Ifallx

n

havethesamedistributionfunctionF

(jr

>

(a),thenthe

existenceofthemathematicalexpectation

E(x)=jadFW(a)

—

oo

isnecessaryandsufficientfor
the

strong

stabilityofs

n

;thesta-

bilityinthis
caseisalwaysnormal

9

.

Again,let

*£l>

X>2)•••

)

X

nt

...

bemutuallyindependentrandomvariables.Thentheprobability

of convergenceoftheseries

fin

(2)

n=l

is equal
eitherto oneor tozero.Inparticular,thisprobability

equalsone whenbothseries

jjEfoJ

and

JSy-fo)

n=l n=l

converge. Letusfurtherassume

y

n

=

x

n

incase[x

n

\^l,

y

n

=

incase

|

x

n

\

>

1.

8

Cf.A.Kolmogorov/Surlaloifortedesgrandesnombres,C.R.Acad.Sci.

Parisv.
191,1930,pp.

910-911.

9

Theproofofthisstatementhasnotyetbeenpublished.