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276 5. Concepts of Stochastic Convergence

the X’s are (i) normal and (ii) non-normal, under appropriate moment
assumptions.
5.3.9 (Exercise 5.2.9 Continued) Suppose that X , ..., X are iid random
1
n
variables having the negative exponential distribution with the common pdf
given by

Let X be the smallest order statistic and . Show that
n:1
the standard exponential random variable, as n → ∞.
5.3.10 Let X , ..., X be iid with the lognormal distribution having the
1
n
following pdf with –∞ < µ < ∞, 0 < σ < ∞:

Denote , the geometric mean of X , ..., X .
1 n
(i) Find the real number c(> 0) such that
(ii) Find the positive real numbers a , b such that (T – a )/b
n n n n n
as n → ∞.
5.3.11 Let X , ..., X be iid with the uniform distribution on the interval (0,
n
1
θ), θ > 0. Denote , the geometric mean of X , ..., X .
1 n
(i) Find c(> 0) such that as n → ∞;
(ii) Find a , b (> 0) such that as n → ∞.
n n
5.3.12 (Exercise 5.2.15 Continued) Suppose that X , ..., X are iid having
1 n
the common N(0, 1) distribution. We denote and
. Find suitable numbers c , d (> 0), c , d (> 0) such
n n n n
that and as n → ∞.
5.3.13 (Exercise 5.2.16 Continued) Suppose that X , ..., X are iid with the
n
1
pdf f(x) = 1/8e –|x|/4 I(x ∈ ℜ) and n ≥ 3. Denote
and . Find suitable numbers
such that
and as n → ∞.
5.3.14 This exercise shows that the converse of the Theorem 5.3.2 is
not necessarily true. Let {U , U; n ≥ 1} be a sequence of random variables
n