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5. Concepts of Stochastic Convergence 273

in terms of the F random variable and ρ. Thus, p can be evaluated with the
2,2
help of the integration of the pdf of F .}
2,2
5.2.15 Suppose that X , ..., X are iid having the common N(0, 1) distribu-
1 n
tion, and let us denote .
Determine the positive real numbers a and b such that as
n → ∞. {Hint: We write

so that which can be rewritten as
by substituting v = 1/2u . Can one of the
2
weak laws of large numbers be applied now?}
5.2.16 Suppose that X , ..., X are iid with the pdf f(x) = 1/8e –|x|/4 I(x ∈ ℜ)
1 n
and n ≥ 3. Denote and
Determine the real numbers a, b and c such that
as n → ∞.
5.2.17 (Example 5.2.11 Continued) Let X , ..., X be iid random variables
1
n
with E(X ) = µ and V(X ) = σ , –∞ < µ < ∞, 0 < σ < ∞, n ≥ 2. Let
2
1 1
be the sample variance. Suppose that Y =
i
aX + b where a(> 0) and b are fixed numbers, i = 1, ..., n. Show that the new
i
sample variance based on the Y ’s is given by
i
5.2.18 Suppose that (X , Y ), ..., (X , Y ) are iid where
1 1 n n
–∞ < µ , µ < ∞, 0 < < ∞ and – 1 < ρ < 1. Let us denote Pearson’s
1
2
sample correlation coefficient defined in (4.6.7) by r instead of r. Show that
n
as n → ∞. {Hint: Use Theorem 5.2.5.}
5.2.19 (Exercise 5.2.11 Continued) Denote T = I n = 1. Does or 1
n
as n →∞? Prove your claim.
5.2.20 Let X , ..., X , ... be iid random variables with the finite variance.
n
1
Let us denote
Show that as n → ∞. {Hint: Verify that for all n.
In this problem, one will need expressions for both Re-
view (1.6.11) as needed.}
5.2.21 Let X , ..., X , ... be iid random variables where
1 n