Page 270 - DMTH404_STATISTICS

Statistics Richa Nandra, Lovely Professional University

Notes Unit 19: The Laws of Large Numbers Compared

CONTENTS

Objectives
Introduction
19.1 Strong Law of Large Numbers
19.2 Summary

19.3 Keywords
19.4 Self Assessment
19.5 Review Questions
19.6 Further Readings

Objectives

After studying this unit, you will be able to:
 Discuss the strong law of large number

 Discuss examples related to large number
Introduction

Probability Theory includes various theorems known as Laws of Large Numbers; for instance,
see [Fel68, Hea71, Ros89]. Usually two major categories are distinguished: Weak Laws versus
Strong Laws. Within these categories there are numerous subtle variants of differing generally.
Also the Central Limit Theorems are often brought up in this context.
Many introductory probability texts treat this topic superficially, and more than once their
vague formulations are misleading or plainly wrong. In this note, we consider a special case to
clarify the relationship between the Weak and Strong Laws. The reason for doing so is that I
have not been able to find a concise formal exposition all in one place. The material presented
here is certainly not new and was gleaned from many sources.

In the following sections, X1, X2, ... is a sequence of independent and indentically distributed
random variabels with finite expectation . We define the associated sequence X of partial
i
sample means by
1 n
i X   X . i
n i 1

The Laws of Large Numbers make statements about the convergence of X to m. Both laws
n
relate bounds on sample size, accuracy of approximation, and degree of confidence. The Weak
Laws deal with limits of probabilities involving X . The Strong Laws deal with probabilities
n
involving limits of X . Especially the mathematical underpinning of the Strong Laws requires
n
a caretful approach ([Hea71, Ch. 5] is an accesible presentation).

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