Page 309 - Elements of Distribution Theory

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

9.8 Exercises 295

and lim r→∞ y r = 0, then

∞ ∞

x j y j = S j (y j − y j+1 ).
j=m j=m
Example 9.18 (Tail probability of the logarithmic series distribution). Consider approx-
imation of the sum
∞
j
θ /j,
j=n
where 0 <θ < 1, as discussed in Example 9.17. We can apply Theorem 9.18, taking y j =
j
1/j and x j = θ . Hence,
1 − θ j−n+1
n
S j = θ , j = n,....
1 − θ
It follows that

∞ j ∞ j−n+1
θ n 1 − θ 1
= θ
j 1 − θ j( j + 1)
j=n j=n
j−n+1
n
∞
∞
θ 1 θ
= − .
1 − θ j( j + 1) j( j + 1)
j=n j=n
Using Corollary 9.3 it is straightforward to show that
∞ ∞
1 1 1
= dx + O
j( j + 1) n x(x + 1) n 2
j=n

1 1 1
= log(1 + 1/n) + O = + O .
n 2 n n 2
Since
∞ j−n+1 ∞ j
θ θ 1

= = O ,
j( j + 1) (n + j − 1)(n + j) n 2
j=n j=1
it follows that
∞ j n
θ θ 1 1
= 1 + O as n →∞.
j 1 − θ n n
j=n
9.8 Exercises

9.1 Prove Theorem 9.4.
9.2 Show that the beta function β(·, ·) satisfies

∞
r−1
t
β(r, s) = s+r dt, r > 0, s > 0.
0 (1 + t)