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264 5. Concepts of Stochastic Convergence

we can immediately conclude that as n → ∞ where Z
is the standard normal variable. !
For the approximation of the Binomial(n, p) distribution with the
N(np, np(1 – p)) distribution, for large n and fixed 0 < p < 1,
refer to the Exercise 5.3.18.

5.4 Convergence of Chi-square, t, and F

Distributions
In this section, we discuss some asymptotic properties of the Chi-square, t,
and F distributions. One will notice that most of the probabilistic and statisti-
cal tools introduced thus far come together in these topics.

5.4.1 The Chi-square Distribution

First, consider a random variable U which is distributed as . Now,
ν
–1
–1
E(ν U ν ) = 1, V(ν U ν) = ν V(U ν) = 2ν → 0 as ν → ∞, and hence by
–1
–2
applying the Weak WLLN, we can claim immediately that
Let X , ..., X ν, ... be iid so that we may view as , and hence we
1
can write . Now, we can apply the CLT (Theo-
rem 5.3.4) to claim that as ν → ∞ with µ = E(X )
1
= 1, σ = V(X ) = 2, and hence, as ν → ∞. In
2
1
the Example 5.3.4, we arrived at the same conclusion with a fairly different
approach. For practical purposes, we would say:

5.4.2 The Student’s t Distribution
The Student’s t random variable was described in the Definition 4.5.1.
Let W ν = X ÷ (Y νν ) where X is the standard normal variable and Y ν is
–1 1/2
, while X and Y ν are assumed independent. In other words, W ν has the