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5. Concepts of Stochastic Convergence 279

some appropriate random variable in distribution as both n and n → ∞ in
2
1
such a way that n /n → k(> 0)?
1 2
5.4.2 Let X , ..., X be iid N(µ , σ ), Y , ..., Y be iid N(µ , 3σ ) where
2
2
1
n
1
1
2
n
–∞ < µ , µ < ∞, 0 < σ < ∞. Also suppose that the X’s and Y’s are
2
1
independent. Denote for n ≥ 2,
(i) Show that is distributed as N (0, 1);
(ii) Show that (n – 1)T /σ is distributed as ;
2
n
(iii) Are V , T independent?
n
n
(iv) Show that is distributed as
the Student’s t ;
2(n–1)
(v) Show that ;
(vi) Show that as n → ∞.
5.4.3 Suppose that X , ..., X are iid exponential with mean β(> 0), Y , ...,
1
1
m
Y are iid exponential with mean η(> 0), and that the X’s are independent of
n
the Y’s. Define T = . Then,
m, n
(i) Show that T is distributed as β/η F ;
m, n 2m, 2n
(ii) Determine the asymptotic distribution of T m, n as n → ∞, when m is
kept fixed;
(iii) Determine the asymptotic distribution of T m, n as m → ∞, when n is
kept fixed.
5.4.4 Verify the limiting ratio of the gamma functions in (5.4.6).
5.4.5 Verify the limiting ratio of the gamma functions in (5.4.14).
5.4.6 Show that the percentile point d ≡ d(ν , α) given by (5.4.17) is a
2
strictly decreasing function of ν whatever be fixed α ∈ (0, 1). {Hint: Take the
2
derivative of d with respect to ν and show that this derivative is negative. This
2
approach is not entirely fair because ν is after all a discrete variable ∈ {1, 2, 3,
2
...}. A rigorous approach should investigate the behavior of d(ν + 1, α) – d(ν ,
2
2
α) for ν ∈ {1, 2, 3, ...}. The “derivative approach” however should drive the
2
point home.}
5.4.7 Consider the random variables T , ..., T defined in (4.6.12) whose
1
p
joint distribution was the multivariate t, denoted by Mt (ν, Σ). Derive the limit-
p
ing distribution of (T , ..., T ) as ν → ∞. Show that the pdf of the Mt (ν, Σ)
1
p
p
distribution given by (4.6.13) converges to the pdf of the corresponding limit-
ing random variable U as ν → ∞. Identify this random variable U by name.
{Hint: In the second part, use techniques similar to those used in Section 5.4.4.}