Page 267 - Probability and Statistical Inference

244 5. Concepts of Stochastic Convergence

Proof For arbitrary but otherwise fixed ε (> 0), we apply the Markov
inequality (Theorem 3.9.1) and write

–r
Thus, we have 0 ≤ P{| T – a |≥ ε} ≤ ξ ε → 0 as n → ∞. That is, P{| T –
r,n
n
n
a |≥ε} ≤ ξ ε → 0 as n → ∞. !
–r
r,n
In Example 5.2.3, the Weak WLLN does not apply,
but Theorem 5.2.2 does.
Example 5.2.3 Let X , ..., X be identically distributed with –∞ < µ < ∞, 0
n
1
< σ < ∞, but instead of assuming independence among these X’s, we only
assume that Cov(X , X ) = 0, for i ≠ j = 1, ..., n, n ≥ 1. Again, we have V( )
j
i
= n σ which converges to zero as n → ∞. Thus, by the Theorem 5.2.2,
2
–1
as n → ∞. !
Example 5.2.3 shows that the conclusion from the Weak WLLN
holds under less restrictive assumption of uncorrelation among
the X’s rather than the independence among those X’s.
Example 5.2.4 Here we directly apply the Theorem 5.2.2. Let X , ..., X
1
be iid Uniform(0, θ) with θ > 0. From Example 4.2.7, recall that T = X , the n
n
n:n
largest order statistic, has the pdf given by
Thus, , and similarly we
get . That is, E{(T – θ) } = 2(n + 1) (n + 2) θ 2
–1
–1
2
n
which converges to zero as n → ∞. Now, we apply the Theorem 5.2.2 with
a = θ and r = 2 to conclude that as n → ∞. !
We can, however, come to the same conclusion without referring to the
Theorem 5.2.2. Look at the next example.
Example 5.2.5 (Example 5.2.4 Continued) Let X , ..., X be iid uniform on
1
n
the interval (0, θ) with θ > 0. Let us directly apply the Definition 5.2.1 to
show that T = as n → ∞. Recall the pdf of T from (5.2.3). Now,
n
n
for arbitrary but otherwise fixed ε > 0, one has

where p is positive. We simply need to evaluate the lim p as n → ∞.
n n
Now, with 0 < ε < θ, we have