Page 297 - Elements of Distribution Theory

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

9.6 Uniform Asymptotic Approximations 283

for some > 0 and, hence,
√ √

∞
exp(x) nφ( nx) dx
z n −1
√ √ = 1 + O(n )as n →∞.
φ( nz n )exp(z n )/( nz n )
However,
√
√ √ exp(z 0 / n) 1 1
φ( nz n )exp(z n )/( nz n ) = φ(z 0 ) = φ(z 0 ) 1 + O √ .
z 0 z 0 n
The exact value of the integral in this case is
1
√ 1
exp{1/(2n)}[1 − (z 0 − 1/ n)] = 1 − (z 0 ) + φ(z 0 )√ + O .
n n
Hence, if the approximation is valid to the order stated, then
1 1
1 − (z 0 ) = φ(z 0 ) 1 + O n − 2 ,
z 0
that is,
1
1 − (z 0 ) = φ(z 0 ) .
z 0
It follows that (9.3) is guaranteed to hold only valid for fixed values of z.

In this section, we present a asymptotic approximation to integrals of the form

∞
√ √
h(x) nφ( nx) dx (9.4)
z
that overcomes both of the drawbacks illustrated in the previous example. Specifically,
this approximation has the properties that the form of the approximation does not depend
on the value of z and that the approximation is valid uniformly in z. The approximation
takes advantage of the fact that the properties of the integral (9.4) when h is a constant are
well-known; in that case,
∞

√ √ √
h(x) nφ( nx) dx = h(z)[1 − ( nz)].
z
Note that it is generally necessary to do a preliminary transformation to put a given integral
into the form (9.4).
Theorem 9.16. Consider an integral of the form
∞

√ √
h n (x) nφ( nx) dx
z
where h 1 , h 2 ,... is a sequence of functions such that
( j)
2
sup |h (x)|≤ c j exp d j x , j = 0, 1, 2
n
n
for some constants c 0 , c 1 , c 2 , d 0 , d 1 , d 2 .
Then, for all M < ∞,
∞

√ √ √ 1 h n (z) − h n (0) √
h n (x) nφ( nx) dx = [1 − ( nz)] h n (0) + O + √ φ( nz),
z n nz
as n →∞, uniformly in z ≤ M.