Page 288 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 265

Student’s t distribution with ν degrees of freedom. Now, since
according to (5.4.1), so does as ν → ∞ by the Theorem 5.2.5
1/2
with g(x) = x , x > 0. Hence, applying Slutsky’s Theorem, we conclude that
as ν → ∞. For practical purposes, we would say:

5.4.3 The F Distribution
–1
The F ν1,ν2 random variable was introduced by the Definition 4.5.2 as ν Xν 1
1

÷ ν Yν where X , Y are independent, X is and Y is . Now, if ν
–1
2 2 ν 1 ν 2 ν 1 ν 2 1
is held fixed, but ν is allowed to go to infinity, then, .
2
Next, we apply Slutsky’s Theorem. Hence, as ν → ∞, we conclude that
2
, that is, for practical purposes, we would say:

5.4.4 Convergence of the PDF and Percentage Points
From (1.7.30), let us recall the pdf of the Student’s t random variable with ν
degrees of freedom, denoted by tν. The pdf of tν, indexed by ν, is given by

with We had
seen in (5.4.3) that , the standard normal variable, as ν → ∞. But, is it
true that fν(x) → φ(x), the pdf of the standard normal variable, for each fixed
x ∈ ℜ, as ν → ∞? The answer follows.
Using the limiting value of the ratio of gamma functions from (1.6.24), we
obtain