Page 311 - Elements of Distribution Theory

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

9.9 Suggestions for Further Reading 297

9.14 Let ψ(·) denote the logarithmic derivative of the gamma function. Show that
1

ψ(z) = log z + O as z →∞.
z
9.15 Let Y denote a real-valued random variable with an absolutely continuous distribution with
density function
c(α)exp{α cos(y)}, −π< y <π,
where α> 0 and c(α)isa normalizing constant; this is a von Mises distribution. Find an
approximation to c(α) that is valid for large values of α.
9.16 Let X denote a real-valued random variable with an absolutely continuous distribution with
density function
1 α−1
(1 − x) , 0 < x < 1.
α
Find an approximation to E[exp(X)] that is valid for large α.
9.17 Let Y denote a real-valued random variable with an absolutely continuous distribution with
density function
√
n 3 n
p(y) = √ y − 2 exp − (y + 1/y − 2) , y > 0;
(2π) 2
this is an inverse gaussian distribution. Find an approximation to Pr(Y ≥ y), y > 0, that is valid
for large n.
9.18 Let Y denote a random variable with an absolutely continuous distribution with density function
1 α−1 α−1
y (1 − y) , 0 < y < 1,
β(α, α)
where α> 0; this is a beta distribution that is symmetric about 1/2. Find an approximation to
Pr(Y ≥ y) that is valid for large α.
9.19 Euler’s constant, generally denoted by γ ,is defined by

1
n
γ = lim − log(n) .
n→∞ j
j=1
Give an expression for γ in terms of the function P defined in Theorem 9.17.
9.20 Prove Theorem 9.18.
9.21 Consider the sum
∞
1
where α> 1;
j α
j=1
this is the zeta function, evaluated at α. Show that
∞
1 1 +
= + O(1) as α → 1 .
j α α − 1
j=1

9.9 Suggestions for Further Reading

The functions described in Section 9.2 are often called special functions and they play an important
role in many fields of science. See Andrews et al. (1999) and Temme (1996) for detailed discussions
of special functions; in particular, Temme (1996, Chapter 11) discusses many special functions that