Page 285 - Elements of Distribution Theory

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

9.4 Watson’s Lemma 271

Hence, using the change-of-variable t =− log(u),
(z) 1 ∞ x−1
= (1 − exp(−t)) exp(−zt) dt
(z + x) (x) 0
and an asymptotic expansion for this integral may be derived using Watson’s lemma.
Note that we may write
x−1
1 − exp(−t)

x−1
x−1
[1 − exp(−t)] = t
t
and, as t → 0,
1 − exp(−t) 1 2
x−1
= 1 − (x − 1)t + O(t );
t 2
hence,
1

x−1 x−1 x x+1
1 − exp(−t) = t − (x − 1)t + O(t )as t → 0.
2
It now follows from Watson’s lemma that
∞ (x) 1 (x + 1) 1

(1 − exp(−t)) x−1 exp(−zt) dt = − (x − 1) + O
0 z x 2 z x+1 z x+2
and, hence, that
(z) 1 1 x(x − 1) 1
= − + O as z →∞.
(z + x) z x 2 z x+1 z x+2
Watson’s lemma may be generalized to allow the function h in
T
h(t)exp{−nt} dt
0
to depend on n.

Theorem 9.13. Let h 1 , h 2 ,... denote a sequence of real-valued continuous functions on
[0, ∞) satisfying the following conditions:
(i) sup h n (t) = O(exp(bt)) as t →∞ for some constant b.
n
(ii) There exist constants c n0 , c n1 ,..., c n(m+1) ,n = 1, 2,...,, a 0 , a 1 ,..., a m+1 ,

−1 < a 0 < a 1 < ··· < a m+1
such that
m
ˆ a j
h nm (t) = c nj t
j=0
satisfies
ˆ
h n (t) = h nm (t) + R nm (t)
where
+
sup R nm (t) = O(t a m+1 ) as t → 0 .
n