Page 307 - Elements of Distribution Theory

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

9.7 Approximation of Sums 293

Proof. Equation (9.10) follows from Theorem 9.17 provided that
r 1

P(x) f (x) dx ≤ | f (r) − f (m)|.
2
m
Consider the case in which f is a decreasing function; the case in which f is increasing
follows from a similar argument.
Since f is decreasing, f (x) ≤ 0on[m,r] and, since |P(x)|≤ 1/2 for all x,

r r

1 1

P(x) f (x) dx ≤− f (x) dx = [ f (m) − f (r)],

2 2
m m
proving (9.10).
Equation (9.11) follows from (9.10) using the fact that

r r r r
1

f ( j) − f ( j) −
2
f (x) dx ≤ f (x) dx − [ f (m) + f (r)]
j=m j=m
m m
1
+ | f (r) − f (m)|
2
and (9.12) follows from (9.10) by taking limits as r →∞.
Example 9.16 (Stirling’s approximation). Consider the function log(n!), n = 0, 1,....
Note that
n

log(n!) = log( j).
j=1
Since log(x)isa strictly increasing function, by Corollary 9.3,
n

log(n!) = log(x) dx + R n = n log(n) − n + 1 + R n
1
where
|R n |≤ log(n)/2.
Hence,

log(n!) = n log(n) − n + O(log(n)) as n →∞,
which is a crude form of Stirling’s approximation. By Example 9.8, we know that the
O(log(n)) term can be expanded as
1 1 −1
log(2π) + log(n) + O(n ).
2 2
Example 9.17 (Tail probability of the logarithmic series distribution). Let X denote a
discrete random variable such that
j
Pr(X = j) = c(θ)θ /j, j = 1, 2,...,
where 0 <θ < 1 and
1
c(θ) =− .
log(1 − θ)