Page 273 - DMTH404_STATISTICS

Unit 19: The Laws of Large Numbers Compared

Notes
Example: Renewal theory. Let X , X , ... be i.i.d. with 0 < X < . Let T = X + ... + X and
1 2 i n 1 n
think of T as the time of nth occurence of some event. For a concrete situation consider a diligent
n
janitor who replaces a light bulb the instant it burns out. Suppose the first bulb is put in at time
0 and let X be the lifetime of the ith lightbulb. In this interpretation T is the time the nth light
i n
bulb burns out and N = sup{n : T  t} is the number of light bulbs that have burns out by
t n
time t.
Theorem. If EX =    then as t  , N /t  1/ a.s. (1/ = 0)
1 t
19.2 Summary

 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 m. We define the associated sequence X of
i
partial sample means by
1 n
i X   X .
i
n i 1

 Lemma. Let Y = K 1 and T = Y + ... + Y . It is sufficient to prove that T /n   a.s.
k k (|X |k) n 1 n n
k
2
 Lemma.   k 1 var(Y )/k  4E|X |  .
1
k

 Lemma. If y  0 then 2y  k y k  2  4.

M
M
M
 Implies S /n  EX . Since X  X it follows that
i
i
i
n
M
lim inf S /n  lim S /n  EX M
n
i
n
n n


M 
M
The monotone convergence theorem implies E(X )  EX   as M, so EX  E(X )
i i i i
M 
 E(X )   and we have lim inf S /n   which imlies the desired result.
i n n
19.3 Keywords
Probability Theory includes various theorems known as Laws of Large Numbers.
Strong law of large numbers. Let X1, X2, ... be pairwise independent identically distributed
random variables with E | X | < . Let EX =  and S = X + ... + X . Then S /n   a.s. as n  .
i i n 1 n n
19.4 Self Assessment
1. .................. includes various theorems known as Laws of Large Numbers.
2. The Laws of Large Numbers make statements about the convergence of .................. to m.
2
3. Lemma.   k 1 var(Y )/k  ..................
k

4. Lemma. If y  0 then ..................
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