Page 284 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 261

Example 5.3.7 First note that we can express

and hence by Slutsky’s Theorem, under the same conditions as the CLT, we

can conclude that This
is so because as n →
∞. Similarly, we can easily handle the convergence in distribution for the
sequence of random variables (i) if µ ≠ 0 or (ii)

if µ > 0 and is positive w.p.1 for all n. These are
left out as the Exercise 5.3.4. !
The following result is a very useful follow-up on the CLT. Suppose that
one is able to use the CLT to verify that converges in distribution
to N(0, σ ) as n → ∞, with some appropriate θ and σ(> 0), where {T ; n ≥ 1}
2
n
is a sequence of real valued random variables. But then, does
converge to an appropriate normal variable for reasonable g(.) functions?
Review the Example 5.3.7. The following theorem, a nice blend of the CLT
(Theorem 5.3.4) and Slutsky’s Theorem, answers this question affirmatively.
Theorem 5.3.5 (Mann-Wald Theorem) Suppose that {T ; n ≥ 1} is a
n
sequence of real valued random variables such that
as n → ∞ where σ (> 0) may also depend on θ. Let g(.) be a continuous real
2
valued function such that g(θ), denoted by g’(θ), is finite and nonzero.
Then, we have:

Proof We proceed along the proof given in Sen and Singer (1993, pp. 231-
232). Observe that

where we denote

Next, note that

and since as n → ∞, by Slutsky’s Theorem, part (ii), we
conclude that as n → ∞. Thus, as n → ∞, by the