Page 288 - Elements of Distribution Theory

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

274 Approximation of Integrals

along with the exact value of this probability and the relative error of the approximation,
for several choices of z and ν. Note that, for a fixed value of z, the relative error of the
approximation decreases as ν increases. However, very high accuracy is achieved only if z
is large as well. In fact, based on the results in this table, a large value of z appears to be at
least as important as a large value of ν in achieving a very accurate approximation.

Consider an integral of the form

n 2

T 1
h(u)exp − u du;
2
−T 0
clearly, this integral may be transformed into one that may be handled by Watson’s lemma
2
by using the change-of-variable t = u /2. This case occurs frequently enough that we give
the result as a corollary below; it is worth noting that the case in which h depends on n can
be handled in a similar manner.

Corollary 9.1. Let h denote a real-valued continuous function on [0, ∞) satisfying the
following conditions:
2
(i) h(t) = O(exp(bt )) as |t|→∞ for some constant b
(ii) there exist constants c 0 , c 1 ,..., c m+1 such that

m
j m+1
h(t) = c j t + O(t ) as t → 0.
j=0
Consider the integral

T 1 n
I n = h(t)exp − t 2 dt
2
−T 0
where T 0 > 0 and T 1 > 0. Then
m
1
c 2 j 2 j+ 2 ( j + 1/2) 1
2
I n = 1 + O m +1 as n →∞.
n j+ 2 n 2
j=0
Proof. First suppose that T 1 = T 0 ≡ T . Note that
T n T n
h(t)exp − t 2 dt = [h(t) + h(−t)] exp − t 2 dt
−T 2 0 2
T 2 √ √
2 h( (2u)) + h(− (2u))

= √ exp{−nu} du
0 (2u)
T 2

2
¯
≡ h(u)exp{−nu} du.
0
Since
m
j m+1
h(t) = c j t + O(t )as t → 0,
j=0