Page 275 - Probability and Statistical Inference

252 5. Concepts of Stochastic Convergence

The variance of each term in (5.2.20) would be equal to
, since (i) we pretend
that µ = 0, and (ii) the X’s are assumed independent. Also, the covariances
such as

and hence from (5.2.20) we get

See (1.6.11) for the sum of successive positive integers. Next, by taking each
covariance term into consideration, similarly one has

Now combining (5.2.17)-(5.2.22), we obtain

so that one writes

Now, obviously as n → ∞. Using the Theorem 5.2.2 with r = 2
and a = σ , it follows that as n → ∞. By applying the Theorem
2
5.2.5, we can also conclude, for example, that as n → ∞ or
as n → ∞.
Remark 5.2.2 If the X’s considered in the Example 5.2.11 were assumed
iid N(µ,σ ), then we would have had µ = 3σ so that (5.2.23) would reduce to
4
2
4
Note that (5.2.24) is also directly verified by observing that is distributed
as so that one can write
.

Example 5.2.12 (Example 5.2.11 Continued) Let us continue working under
the non-normal case and define where 0 < γ < 1. Now, by
the Markov inequality and (5.2.23), with k = µ – (n – 3)(n – 1) σ and for
–1
4
4
n
any fixed ε(> 0), we can write:
Hence, by the Definition 5.2.1, as n → ∞, whenever µ is finite and
4
4
µ > σ . !
4