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280 5. Concepts of Stochastic Convergence

5.4.8 Consider the random variables U , ..., U defined in (4.6.15) whose
p
1
joint distribution was the multivariate F, denoted by MF (ν , ν , ..., ν ). De-
0
p
p
1
rive the limiting distribution of (U , ..., U ) as ν → ∞ but ν , ..., ν are held
p
0
1
p
1
fixed. Show that the pdf of the MF (ν , ν , ..., ν ) distribution given by (4.6.16)
0
p
p
1
converges to the pdf of the corresponding limiting random variable W as ν 0
→ ∞ but ν , ..., ν are held fixed. Identify this random variable W by name.
1
p
{Hint: In the second part, use techniques similar to those used in Section
5.4.4.}
5.4.9 (Exercise 5.4.1 Continued) Suppose that the random vari-
ables X , ..., X are iid N(µ , σ ), i = 1, 2, and that the X ’s are
2
in
i1
i
1j
independent of the X ’s. With n ≥ 2, let us denote
2j
for i = 1, 2. Consider the random variable
1, 2. Show that (T , T ) has an appropriate bivariate t distribution,
1n
2n
for all fixed n ≥ 2. Find the limiting distribution of (T , T ) as n →
2n
1n
∞.
5.4.10 (Exercise 5.4.9 Continued) Suppose that the random vari-
ables X , ..., X are iid N(µ , σ ), i = 1, ..., 4, and that the X ’s are
2
ij
i1
in
i
independent of the X ’s for all i ≠ l = 1, ..., 4. With n ≥ 2, let us
lj
denote
for i = 1, ..., 4. Consider the random variable
1, ..., 4. Show that (T , ..., T ) has an appropriate four-dimensional t
4n
1n
distribution, for all fixed n ≥ 2. Find the limiting distribution of (T , ...,
1n
T ) as n → ∞.
4n
5.4.11 (Exercise 5.4.10 Continued) Suppose that the random vari-
ables X , ..., X are iid N(µ , σ ), i = 1, ..., 4, and that the X ’s are
2
inz
ij
i
i1
independent of the X ’s for all i ≠ l = 1, ..., 4. With n ≥ 2, let us
i
lj
denote
for i = 1, ..., 4. Suppose that n = n = n = k and n = n. Consider the
2
1
3
4
random variable . Show that (T , T , T ) has
3n
1n
2n
an appropriate three-dimensional F distribution, for all fixed n ≥ 2.
Find the limiting distribution of (T , T , T ) as n → ∞.
1n 2n 3n