Page 306 - Elements of Distribution Theory

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

292 Approximation of Integrals

for any constant c,
1 1

2 u + j + 1/2 2 u + j + 1/2
uf
du = u f
1 m 1 m
− −
2 2

j + 1/2
− f
du, j = 0,..., m − 1.
m
Using the Lipschitz condition on f

1 1

2 u + j + 1/2 K 2 K
uf du ≤ |u| du = , j = 0,..., m − 1

2

1 m m 1 12m

− −
2 2
so that
m
K

(x − x − 1/2) f (x/m) dx ≤

12
0
and, hence, that
1 m 1

(x − x − 1/2) f (x/m) dx = O as m →∞.
m(m + 1) 0 m 2
It follows that

X m
E f = E[ f (U)]
m m + 1
1 1
+ [ f (0) + f (1)] + O as m →∞.
2(m + 1) m 2
In order for the result in Theorem 9.17 to be useful, we need some information regarding
the magnitude of the integral (9.9), as in the previous example. A particularly simple bound
is available for the case in which f is a monotone function.
Corollary 9.3. Let f denote a continuously differentiable monotone function on [m,r]
where m and r are integers, m ≤ r. Then

r r
1 1
f ( j) − f (x) dx − [ f (m) + f (r)] ≤ | f (r) − f (m)| (9.10)

m 2 2
j=m
and

r r

f ( j) − f (x) dx ≤ max{| f (m)|, | f (r)|}. (9.11)

j=m
m
Suppose that f is a decreasing function and
lim f (r) = 0.
r→∞
Then

∞ ∞
1 1

f ( j) − f (x) dx − f (m) ≤ | f (m)|. (9.12)

m 2 2
j=m