Theory of Statistics - George Mason University

1.4 Limit Theorems 103
The bn can be chosen so that bn ≤ n, as bn = E
XiI{|Xi|≤n} , for example.
Theorem 1.52 (SLLN for iid random variables)
Let X1, X2, . . . be a sequence of iid random variables such that ∀i E(|Xi|) =
µ < ∞. Then
A slight generalization is the alternate conclusion
1
n
n
i=1
1
n Sn
a.s.
→ µ. (1.209)
ai(Xi − E(X1)) a.s.
→ 0,
for any bounded sequence of real numbers a1, a2, . . ..
We can generalize these two limit theorems to the case of independence
but not necessarily identical distributions, by putting limits on normalized p th
moments.
Theorem 1.53 (WLLN for independent random variables)
Let X1, X2, . . . be a sequence of independent random variables such for some
constant p ∈ [1, 2],
1
lim
n→∞ np n
E(|Xi| p ) = 0.
Then
1
Sn −
n
i=1
n
p
E(Xi) → 0. (1.210)
i=1
Theorem 1.54 (SLLN for independent random variables)
Let X1, X2, . . . be a sequence of independent random variables such for some
constant p ∈ [1, 2],
Then
∞
i=1
1
Sn −
n
E(|Xi| p )
i p
i=1
< ∞.
n
a.s.
E(Xi) → 0. (1.211)
We notice that the normalizing term in all of the laws of large numbers
has been n −1 . We recall that the normalizing term in the simple central limit
theorem 1.38 (and in the central limit theorems we will consider in the next
section) is n −1/2 . Since the central limit theorems give convergence to a nondegenerate
distribution, when the normalizing factor is as small as n −1/2 , we
cannot expect convergence in probability, and so certainly not almost sure convergence.
We might ask if there is some sequence an with n −1/2 < an < n −1 ,
such that when an is used as a normalizing factor, we have convergence in
Theory of Statistics c○2000–2013 James E. Gentle