Page 272 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 249

which → 0 as n → ∞. The proof of part (iii) is now complete. !
Convergence in probability property is closed under the
operations: addition, subtraction, multiplication and division.
Caution: Division by 0 or ∞ is not allowed.

Example 5.2.7 Suppose that X , ..., X are iid N(µ, σ ), –∞ < µ < ∞, 0 < σ < ∞,
2
2
1 n
n ≥ 2. Let us consider the sample mean and the sample variance . We know
that as n → ∞. Thus by the Theorem 5.2.4, part (i), we
conclude that as n → ∞, and as n → ∞.!
Let us again apply the Theorem 5.2.4. Suppose that as n → ∞.
Then, by the Theorem 5.2.4, part (i), we can obviously claim, for example,
that as n → ∞. Then, one may write

in view of the Theorem 5.2.4, part (ii). That is, one can conclude:
as n → ∞. On the other hand, one could alternatively think of
where V = U and directly apply the Theorem 5.2.4, part (ii) also. The fol-
n
n
lowing theorem gives a more general result.
Theorem 5.2.5 Suppose that we have a sequence of real valued random
variables {U ; n = 1} and that as n → ∞. Let g(.) be a real valued
n
continuous function. Then, as n → ∞.
Proof A function g(x) is continuous at x = u provided the following holds.
Given arbitrary but otherwise fixed ε(> 0), there exists some positive number
δ → δ(ε) for which

Hence for large enough n(≥ n ≡ n (ε)), we can write
0 0

and the upper bound → 0 as n → ∞. Now, the proof is complete. !
Example 5.2.8 (Example 5.2.4 Continued) Let X , ..., X be iid uniform on
1
n
the interval (0, θ) with θ > 0. Recall from the Example 5.2.4 that T = X ,
n
n:n
the largest order statistic, converges in probability to θ. In view of the
Theorem 5.2.5, obviously as n → ∞, by considering the