Page 458 - A First Course In Stochastic Models

C. GENERATING FUNCTIONS 453

region |z| < R in the complex plane for some R > 1. It is essential that the radius
R is larger than 1. Note that the generating function (C.5) is indeed of the form
(C.6), where the numerator and denominator are analytic functions on the whole
complex plane (R = ∞). It is no restriction to assume that N(z) and D(z) have
no common zeros; otherwise, cancel out common zeros. Let us further assume that
the following regularity conditions are satisfied:

C1 The equation D(z) = 0 has a real root z 0 on the interval (1,R).

C2 The function D(z) has no zeros in the domain 1 < |z| < z 0 of the complex
plane.
C3 The zero z = z 0 of D(z) is of multiplicity 1 and is the only zero of D(z) on
the circle |z| = z 0 .

The following theorem is of utmost importance. The insightful proof of the
theorem is included for completeness. Recall that f (x) ∼ g(x) as x → ∞ means
that f (x)/g(x) → 1 as x → ∞.

Theorem C.1 Under the conditions C1 to C3,
−j
p j ∼ γ 0 z 0 as j → ∞, (C.7)

where the constant γ 0 is given by
1 N(z 0 )
γ 0 = − . (C.8)
z 0 D (z 0 )
′
′
Here D (z 0 ) denotes the derivative of D(x) at x = z 0 .
Proof We first mention the following basic facts from complex function theory.
The most important fact is that a function f (z) is analytic at a point z = a if
∞
a

n
and only if f (z) can be expanded in a power series f (z) = n=0 n (z − a) in
|z − a| < ρ for some ρ > 0. The coefficient a n of the Taylor series is the nth
derivative of f (z) at z = a divided by n!. The analytic function f (z) is said to
have a zero of multiplicity k in z = a if a 0 = · · · = a k−1 = 0 and a k = 0. Another
basic result is the following. The Taylor series
∞ a n
n=0 n (z−a) of a function f (z)
at the point z = a coincides with the function f (z) in the interior of the largest
circle whose interior lies wholly within the domain on which f (z) is analytic.
The proof of (C.7) now proceeds as follows. The conditions C1 to C3 imply
that there is a circle around z = 0 with radius R 0 larger than z 0 such that P (z) is
analytic in |z| < R 0 except for the isolated point z = z 0 . Since D(z) has a zero of
multiplicity 1 at z = z 0 , it follows from the Taylor series that D(z) = (z − z 0 )φ(z)
in |z| < R 0 , where φ(z) is an analytic function with φ(z 0 ) = 0. Thus we can write
P (z) as P (z) = H(z)/(z − z 0 ) for some analytic function H(z) in |z| < R 0 with
H(z 0 ) = 0. Using a Taylor expansion H(z) = H(z 0 ) + (z − z 0 )U(z), we next find