Page 297 - Probability and Statistical Inference

274 5. Concepts of Stochastic Convergence

Does as n → ∞ for some appropriate c? {Hint: Can one of the weak
laws of large numbers be applied here? Is it true that a log(r) = r log(a) for any two
positive numbers a and r?}
5.2.22 (i) (Monotone Convergence Theorem) Consider {U ; n = 1}, a
n
sequence of real valued random variables. Suppose that as as n → ∞.
Let g(x), x ∈ ℜ, be an increasing real valued function. Then, E[g(U )] → g(u)
n
as n → ∞.
(ii) (Dominated Convergence Theorem) Consider {U ; n ≥ 1}, a se-
n
quence of real valued random variables. Suppose that as n → ∞. Let
g(x), x ∈ ℜ, be a real valued function such that |g(U )| ≤ W and E[W] is finite.
n
Then, E[g(U )] → g(u) as n → ∞.
n
{Note: Proofs of these two results are beyond the scope of this book. The
part (i) is called the Monotone Convergence Theorem, a special form of which
was stated earlier in Exercise 2.2.24. The part (ii) is called the Dominated
Convergence Theorem. The usefulness of these two results is emphasized in
Exercise 5.2.23.}
5.2.23 Let X , ..., X be iid non-negative random variables with µ ∈ ℜ , σ
+
1
n
∈ ℜ . We denote .
+
(i) Does E{e } converge to some real number as n → ∞?
U
n
(ii) Does E{e –2Un } converge to some real number as n → ∞?
(iii) Does E{sin(U )} converge to some real number as n → ∞?
n
(iv) Does E{Φ(U )} converge to some real number as n → ∞ where
n
(v) Does E(T ) converge to some real number as n → ∞?
n
{Hints: By the WLLN, we first claim that µ as n → ∞. Let g(x) = e ,
x
x ∈ ℜ be our increasing real valued function. Then, by the Exercise 5.2.22,
+
µ
U
part (i), we conclude that E[g(U )] = E{e } → e as n → ∞. For the second
n
n
part, note that g(x) = e , x ∈ ℜ is bounded between two fixed numbers, zero
+
–2x
+
and one. For the third part, note that g(x) = sin(x), x ∈ ℜ is bounded between
± 1. Thus, by the Exercise 5.2.22, part (ii), we conclude that E[g(U )] will
n
–2µ
converge to e (or sin (µ)) as n → ∞ in part (ii) (or (iii)) respectively. How
about parts (iv)-(v)?}
5.2.24 Let X , ..., X be iid Poisson (λ), 0 < λ < ∞. We denote U =
n
n
1
λ
, n ≥ 1. From Exercise 5.2.23, it follows that E[eU ] → e and E[e –2U n ]
n
–2λ
→ e as n → ∞. Verify these two limiting results directly by using the
mgf of a Poisson random variable. {Hint: Observe that e = 1 + t/n + 1/
t/n
tU
2
n
t/n
2!(t/n) + ... . Then, note that E[e ] = [exp{– λ + λe }] = exp{–nλ +
n