NATIONAL BUREAU OF STANDARDS REPORT

NBS Project

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I

“£>0 “*0002 5 November 1951

NBS Report

1382

A USEFUL FORM THE STRONG LAW OF LARGE TVTT T T7»T^

By

Kal Lai Chvrg

<NBS>

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

This report is issued

production in any forrr writing is obtained fr Washington 25, D. C.

Approved for public release by the

Director of the National Institute of

Standards and Technology (NIST) on October 9, 2015

on, reprinting, or re- ed unless permission in

1 Bureau of Standards,
1
FOREWORD

In this report the Strong Law of Large Numbers is stated

and proved in a form that has been found to be particularly

r useful in connection *&th work on the mathematical theory of

diract^measurement processes currently in progress in the

Statistical Engineering Laboratory of the National Bureau

of Standards 0

The formulation of the Strong few here presented „ and

its proof, were accomplished originally by Professor So Do

Chung while he was a guest worker at the National Bureau of

Standards during the summer of 1951 o This work was performed

under Contract CST~525 between the National Bureau of Standards

and the University of North Carolina e

’ Subsequently Professor

Chung submitted a somewhat shorter proof that had boon

communicated to him by Professor William Feller of Princeton

University, and it is this more concise proof that is reproduced

in this i

5 © porto

J* E 0 Curtiss

Chief, National Applied

Mathematics Laboratories

A * V o Astin

Acting Director

National Bureau of Standards
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A USEFUL FORM OF THE STRONG LAW OF LARGE NUMBERS

by

Kai~Lai Chung

Theorems Let {X } be a g@qiu©no© of in&@p@nd©ntp identically

distributed random variables with mean O 0 Let {o^} be a

hounded sequence of real numbers:

In the special oae© where all e

p

s 1, this la the well®

known Kolmogorov 8 s strong law of large numbers ([1] p 0 §9) 0

Proof:* Define

Then w© have

* This very simple proof was communicated to m© by Professor

Feller; my original proof was unnecessarily longer*
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$©3ct, the series

09 0a

x^ - 0a

E{ 2^}

n

converges with probability one, by Kolmogorov 9 s criterion

{loe* ©lt «) 9 for

1 is ««&2

> < °'Zr5 J *

2 ">*>

n»l n”i a

|x|<n

* C s r ~2 £ j x'"’ <LP(x)

. n^L a k„i k=1<ix!<k

< C

s f * J |x|d»(x) f i

k“l k»l<lx|<fc n=k a

< 2G

2 £ J

|x|dP(x) <».

k"1

fc~l<j|xj< k

It follows as usual that

n

(“» &

^- eu*’

, <,) 1

H©no© also n

pflim

n«*00

k

2

x

CkUk ^E(Xk

))

0 ] S 1

n

Slno© E(Xk ) ® 0, clearly lim E|X }

“ k^a> *

» 0 and eo

lim

J.Ok E(ll) k*l s is n

Thus the theorem follows
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k 0

: 1C •* -

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Corollary It Let } 9 { o } p {X n

)b© altogether independent

random variables such that the variables in each set are identically

distributed o Suppose that the means all exist and are equal t©

Eix) „ E(a) 9 E(x) respectively o Furthermore suppose that the

o

9 s are uniformly bounded? |an | J

K with probability ons 0

Consider th© sequence {Yn

}s

ork

X

nk^ ! * V 0 ° *

o where n

1< 000 < n^,. eoJ> and n - n

k k«r

< M0 Then the strong

law of large numbers holds for {Yn} as follows

) B 1 o

Corollary 2% As in Corollary 1 # consider the sequence
* £

sGq

'*< ,v-

- *

• 3 r - r» o>

~ ^ a

{ F* : £ «* - - • ....' "-SIT"""»« » — r fl*

r>on«iJp-.>i arid naMafloo ,1 V^XXoftoC n. • . s.'.Ic7< .

a * . « • x*o- caXYra , . 1+00* !,,> I IB • •

« BV»

x

fl St*

o c © + x+_n X’ t0«0®® 80 ®
Jr

a Y

1 nk-l +1

b o e o $ a Z

1 n, * he >j. k-1

* ° • ® * *

where the s are re&l eonstantg p |a4 j < A 0 Then the strong

law of large numbers holds for* {^n jas follows i

4

r lira

P I Z1 + Zn

u *«=?> OO n ( V«

.+aa) (B; c) E( X) EfcJ})

[1] A 0 H 0 Kolmogorov, aruu&b©griffe d@? WahrsGheinXi©hk©its«

r©ohausigs Chalsea Publishing 0o o5 Hew York,

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