Page 299 - Elements of Distribution Theory

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

9.6 Uniform Asymptotic Approximations 285

and
3 3

2
sup |g (x)|≤ sup |h (x)|≤ c 2 exp d 2 x
n n
n 2 n 2
and

2
sup |g (x) − 1|≤ c 3 exp d 3 x

n
n
for some constants c 2 , d 2 , c 3 , d 3 .It follows that
∞ ∞ ∞

√ √ √ √ √ √

g (x) nφ( nx) dx − nφ( nx)dx ≤ c 3 nφ( (n − 2d 3 )x) dx.
n
z z z
Note that
∞

√ √ 1 √
nφ( (n − 2d 3 )x) dx = √ [1 − ( (n − 2d 3 )z)].
z (1 − 2d 3 /n)
Hence,
1 ∞ √ √ √ 1
g (x) nφ( nx) dx = [1 − ( nz)] [1 + R n (z)]

n
n z n
where
√
1 1 − ( (n − 2d 3 )z)
|R n (z)|≤ c 3 √ √ .
(1 − 2d 3 /n) 1 − ( nz)
The result follows provided that, for any M < ∞,
sup |R n (z)|= O(1) as n →∞.
z≤M
√
First note that, for z < 1/ n
1 1
|R n (z)|≤ c 3 √ .
(1 − 2d 3 /n) 1 − (1)
By Theorem 9.11,
√ 1 1 √
1 − (( (n − 2d 3 )z) ≤ √ φ( (n − 2d 3 )z)
(n − 2d 3 ) z
and
√ 2
√ nz 1 √
1 − ( nz) ≥ φ( nz).
2
1 + nz z
√
Hence, for z ≥ 1/ n,
2c 3
2
|R n (z)|≤ exp d 3 z .
1 − 2d 3 /n
It follows that
1 1 2c 3
2

sup |R n (z)|≤ max c 3 √ , exp d 3 M ,
z≤M (1 − 2d 3 /n) 1 − (1) 1 − 2d 3 /n
proving the result.