Page 274 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 251

Remark 5.2.1 Under the setup of the Example 5.2.10, using the Theo-
rems 5.2.4-5.2.5, it is obvious that 0 as n → ∞. But note
that the term n → ∞ as n → ∞ and the term V is inflated by this growth
γ
n
factor. It is noteworthy in the Example 5.2.10 that the inflated random vari-
able as n → ∞ if 0 < γ < 1 is held fixed.
Example 5.2.11 Let X , ..., X be iid random variables with E(X ) = µ and
1 n 1
2
V(X ) = σ , – ∞ < µ < ∞, 0 < σ < ∞, n ≥ 2. As usual, write for the
1
sample mean and variance respectively. It is easy to see that

which reduces to (n – 1)σ so that , without assuming any spe-
2
cial distribution. But, the calculation of is not so simple when the distri-
bution of the X’s is unspecified. In the Section 2.3, we had defined the central
moments. In particular, we denoted the third and fourth central moments by
respectively. Assume that 0 < µ <
4
∞ and µ > σ . Since does not depend on the specific value of µ,
4
4
without any loss of generality, let us pretend from this point onward that µ =
0. Look at the Exercise 5.2.17. Now,

which implies that

But, the X’s are iid and hence

and also,