Page 308 - Elements of Distribution Theory

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

294 Approximation of Integrals

This is known as a logarithmic series distribution. Consider approximation of the tail
probability
∞
j
c(θ) θ /j
j=n
for large n.
x
Note that θ /x is a decreasing function so that by Corollary 9.3,
∞ ∞
j x
1
θ /j = θ dx + R n
j=n n x
where
1 n
|R n |≤ θ /n.
2
We may write
∞ θ ∞ 1
n
x −1 x exp{nu} du;
θ dx =
n log(θ) 0 1 + u/ log(θ)
hence, by Watson’s lemma,
∞ θ 1 1
n
x −1 x + O 2 .
θ dx =
n log(θ) n n
Therefore, based on this approach, the magnitude of the remainder term R n is the same as
n
that of the integral itself, θ /n. Hence, all we can conclude is that
∞
j n 1
θ /j = θ O as n →∞.
n
j=n
The previous example shows that the approximations given in Corollary 9.3 are not very
useful whenever the magnitude of | f (r) − f (m)| is of the same order as the magnitude of the
sum f (m) + ··· + f (r). This can occur whenever the terms in the sum increase or decrease
very rapidly; then f tends to be large and, hence, the remainder terms in Corollary 9.3 tend

to be large. In some of these cases, summation-by-parts may be used to create an equivalent
sum whose terms vary more slowly. This result is given in the following theorem; the proof
is left as an exercise.

Theorem 9.18. Consider sequences x 1 , x 2 ,... and y 1 , y 2 ,.... Let m and r denote integers
such that m ≤ r. Define
S j = x m +· · · + x j , j = m,...,r.

Then
r r−1

x j y j = S j (y j − y j+1 ) + S r y r .
j=m j=m
If
∞ ∞

|x j y j | < ∞, |x j | < ∞
j=m j=m