Page 294 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

280 Approximation of Integrals

Theorem 9.15. Consider the integral

b
I n = h(y)exp{ng(y)} dy, −∞ < a < b ≤∞
a
where
(i) g is twice differentiable on [a, b)
(ii) h is differentiable on [a, b)
(iii) g is maximized at y = a
(iv) g (y) < 0 for all a ≤ y < b and h(a) = 0.

Then
h(a) 1 −1
I n = exp{ng(a)} [1 + O(n )] as n →∞.
−g (a) n

Proof. Note that
b b
h(y)exp{ng(y)} dy = exp{ng(a)} h(y)exp{−n[g(a) − g(y)]} dy.
a a
Consider the change-of-variable u ≡ u(y) = g(a) − g(y); note that, since g is strictly
decreasing, u is a one-to-one function of y. Also note that u(a) = 0.
It follows that
b u(b)
h(y(u))
h(y)exp{−n[g(a) − g(y)]} dy = exp{−nu} du

a 0 g (y(u))
u(b)

¯
≡ h(u)exp{−nu} du.
0
Under the conditions on h and g,
¯
¯
h(u) = h(0) + O(u)as u → 0
and, hence, by Watson’s lemma,
u(b) 1
¯
¯
−2
h(u)exp{−nu} du = h(0) + O(n )as n →∞.
u(a) n
The result now follows from the fact that
h(a)
¯
h(0) = .
−g (a)

Example 9.11 (Pareto distribution). Let Y denote a real-valued random variable with an
absolutely continuous distribution with density
α
p(y; α) = , y ≥ 1;
y α+1
here α> 0isa parameter. This is a Pareto distribution. Consider an approximation to
2
E[log(1 + Y ); α] for large values of α.
We may write
∞ α ∞ log(1 + y )
2
2 2
E[log(1 + Y ); α] = log(1 + y ) α+1 dy = α exp{−α log(y)} dy.
1 y 1 y