Page 280 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 257

Theorem 5.3.3 (Slutsky’s Theorem) Let us consider two sequences of
real valued random variables {U , V ; n ≥ 1}, another real valued random
n n
variable U, and a fixed real number v. Suppose that v as
n → ∞. Then, we have as n → ∞:
(i)
(ii)

(iii) provided that P{V = 0} = 0 for all n
n
and v ≠ 0.

Example 5.3.5 (Examples 5.3.1-5.3.2 Continued) Recall T = X in the
n n:n
Uniform(0, θ) situation and let us define . We already
know that . By using the Theorem 5.2.4 we can conclude
that . We had proved that U as n → ∞
where U has the standard exponential distribution. Hence, by Slutsky’s Theo-
rem, we can immediately claim that as n → ∞. We
can also claim that as n → ∞. It is not easy to
obtain the df of H and then proceed directly with the Definition 5.3.1 to show
n
that . !

In the Exercise 5.3.2, we had suggested a direct approach using
the Definition 5.3.1 to show that

5.3.2 The Central Limit Theorems

First we discuss the central limit theorems for the standardized sample mean
and the sample variance.

Let X , ..., X be iid random samples from a population with
1 n
mean µ and variance σ , n = 2. Consider the sample mean
2
and the sample variance . In the literature,
is called the standardized version of the sample mean when
σ is known, and is called the standardized
version of the sample mean when σ is unknown.

The following theorem, known as the central limit theorem (CLT),
provides under generality the asymptotic distribution for the standardized