Page 263 - DMTH404_STATISTICS

Unit 18: The Weak Law

A stronger version of this result due to Borel and Cantellistates that the above ratio k/n tends to Notes
p not only in probability, but with probability 1. This is the strong law of large numbers (SLLN).
What is the difference between the weak law and the strong law? The strong law of large
numbers states that if { } is a sequence of positive numbers converging to zero, then
n
  k 
 P   p   n    ...(18.2)
n 1  h 

  k 
From Borel-Cantelli lemma [see (2-69) Text], when (13-2) is satisfied the events An    p   n
 h 
can occur only for a finitenumber of indices n in an infinite sequence, or equivalently, the events
 k 
  p   n  occur infinitely often, i.e., the event k/nconverges to palmost-surely.
 h 

Proof: To prove (18.2), we proceed as follows. Since
k
4
 p    |k – np |   n 4
4
h
we have

n   k   k  
 (k np) p (k) 4 n   4 n  P   p     p   
4
4
4
P


n

k 0   n   n 
and hence
n
4
 (k np) p (k)

 k  n


P   p    k 0 4 4 ...(13.3)
 n   n
where
 n   n  k n k

p (k) P  X     p q
k 

n i
k
 i 1   

By direct computation
4
4


n    n      n  
 (k np)4pn(k) E  X  np   E  X  p 




i
i



k 0    i 1       i 1   
4
   n    n n n n

 E  Y i    E(YY YY )
k
l
j
i
   i 1    i 1 k 1 j 1 l 1





n n n n n
3
2
4
2
  E(Y ) 4n(n 1)  E(Y )E( Y ) 3n(n 1)  E(Y )E(Y )




i
j
i
i
j

i 1 i 1 j 1 i 1 j 1




2
= n(p + q )pq + 3n(n – 1)(pq)  [n + 3n(n – 1)]pq
3
3
= 3n pq, ...(18.4)
2
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