Page 268 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 245

–1
n
(1 – ε/θ) → 0 as n → ∞, since 0 < 1 – εθ < 1. Hence, by the Definition
5.2.1, we can claim that as n → ∞.
Weak WLLN (Theorem 5.2.1) can be strengthened considerably.
One claims: as n → ∞ so long as the X ’s are
i
iid with finite µ. Assuming the finiteness of σ is not essential.
2
In what follows, we state a stronger version of the weak law of large
numbers. Its proof is involved and hence it is omitted. Some references are
given in the Exercise 5.2.2.
Theorem 5.2.3 (Khinchine’s WLLN) Let X , ..., X be iid real valued
1 n
random variables with E(X ) = µ, –∞ < µ < ∞. Then, as n → ∞.
1
Example 5.2.6 In order to appreciate the importance of Khinchine’s WLLN
(Theorem 5.2.3), let us consider a sequence of iid random variables {X ; n =
n
1} where the distribution of X is given as follows:
1

with so that 0 < c < ∞. This is indeed a probability distribution.
Review the Examples 2.3.1-2.3.2 as needed. We have which
is finite. However, , but this infinite series is not finite. Thus,
we have a situation where µ is finite but ó is not finite for the sequence of the
2
iid X’s. Now, in view of Khinchine’s WLLN, we conclude that as
n → ∞. From the Weak WLLN, we would not be able to
reach this conclusion.
We now move to discuss other important aspects. It may be that a se-
quence of random variables U converges to u in probability as n → ∞, but
n
then one should not take it for granted that E(U ) → u as n → ∞. On the other
n
hand, one may be able to verify that E(U ) → u as n → ∞ in a problem, but
n
then one should not jump to conclude that as n → ∞. To drive the
point home, we first look at the following sequence of random variables:

It is simple enough to see that E(U ) = 2 – 1/n → 2 as n → ∞, but as
n
n → ∞. Next, look at a sequence of random variables V :
n