Page 292 - Probability and Statistical Inference

5. Concepts of Stochastic Convergence 269

In the same vein, when ν = 2, but ν → ∞, we can easily study the behav-
1 2
ior of F ν in terms of for any fixed 0 < α < 1. We rely upon very simple
2, 2,α
and familiar tools for this purpose. In this special case, we provide the interest-
ing details so that we can reinforce the techniques developed in this textbook.
Let us denote c for the upper 100α% point of the distribution of ,
that is

and hence we have c = c(α) = –log(α). Suppose that X and Y are respectively
distributed as and , and suppose also that they are independent. Let h(y)
be the pdf of Y at y(> 0). Let us simply write d instead of F ν so that
2, 2,α

Since X and Y are independent, one obtains

writing the mgf of from (2.3.28). Thus, one has the following exact
relationship between d and α:

One can show that d is a strictly decreasing function of ν . We leave this out
2
as an exercise. Next, let us explore how we can expand d as a function of ν .
2
We use a simple trick. Observe that