Page 298 - Elements of Distribution Theory

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

284 Approximation of Integrals

Proof. We may write
∞ ∞

√ √ √ √
h n (x) nφ( nx) dx = h n (0) nφ( nx) dx
z z
∞

h n (x) − h n (0)√ √
+ x nφ( nx) dx
z x
∞
√ h n (x) − h n (0)√ √
= h n (0)[1 − ( nz)] + x nφ( nx) dx.
z x
Let
h n (x) − h n (0)
g n (x) = .
x
Note that
√ √ d √
nxφ( nx) =− φ( nx);
dx
hence, using integration-by-parts,
∞ ∞

√ √ 1 √ √ ∞ 1 √ √

g n (x) nxφ( nx) dx =− g n (x) nφ( nx) + g (x) nφ( nx) dx
n
z n z n z
√
φ( nz) 1 ∞ √ √

= g n (z) √ + g (x) nφ( nx) dx.
n
n n z
It follows that
√
∞

√ √ √ φ( nz)
h n (x) nφ( nx) dx = h n (0)[1 − ( nz)] + g n (z) √
n
z
1 ∞ √ √

+ g (x) nφ( nx) dx.
n
n z
Hence, the theorem holds provided that
1 ∞ √ √ √
1
g (x) nφ( nx) dx = [1 − ( nz)]O ,

n
n z n
where the O(1/n) term holds uniformly for z ≤ M, for any M < ∞.
Note that

h (x) h n (x) − h n (0) h n (x) − h n (0) − xh (x)

n n
g (x) = − =− .
n
x x 2 x 2
Using Taylor’s series approximations,
1 2
h n (x) = h n (0) + h (0) + h (x 1 )x

n n
2
and
h (x) = h (0) + h (x 2 )x

n n n
where |x j |≤|x|, j = 1, 2. Hence,
1

g (x) = h (x 1 ) − h (x 2 )
n n n
2