Page 459 - A First Course In Stochastic Models

454 APPENDICES

that P (z) can be represented as
r 0
P (z) = + U(z) (C.9)
z − z 0
in |z| < R 0 , z = z 0 . Here U(z) is an analytic function in the domain |z| < R 0 and
the residue r 0 = H(z 0 ) is given by

′
r 0 = lim (z − z 0 )P (z) = N(z 0 )/D (z 0 ).
z→z 0
The remainder of the proof is simple. Since U(z) is analytic for |z| < R 0 we have

∞ j
u
the power series representation U(z) = j=0 j z for |z| < R 0 . Let R 1 be any
! ! −j
number with z 0 < R 1 < R 0 . Then, for some constant b, u j ≤ bR for all j ≥ 0.
! !
1
j
This follows from the fact that the series
j=0 j z is convergent for z = R 1 .
∞
u
Using the power series representation of U(z) and the fact that the power series

j
∞
representation P (z) = p j z extends to |z| < z 0 , it follows from (C.9) that
j=0
∞ ∞ ∞
j j j
−r 0
p j z = (z/z 0 ) + u j z , |z| < z 0 .
z 0
j=0 j=0 j=0
Equating coefficients yields
−j−1
p j = −r 0 z + u j , j ≥ 0.
0
! ! −j
Since u j ≤ bR for some constant b and R 1 > z 0 , the coefficient u j tends to
1
! !
−j
zero faster than z . Hence we can conclude the asymptotic expansion (C.7).
0
It is noted that Theorem C.1 does not require that {p j } is a probability distri-
bution. The theorem applies to any sequence {p j , j = 0, 1, . . . } with p j ≥ 0 for
all j and
j=0 j < ∞. The asymptotic expansion (C.7) is very useful for both
∞
p
theoretical and computational purposes. It appears that in many applications the
asymptotic expansion for p j can be used for relatively small values of j. To illus-
trate this, consider the generating function (C.5) for the problem of success runs.
s s
This generating function P (z) is the ratio of the two analytic functions N(z) = p z

s k−1 k
and D(z) = 1− p (1−p)z whose domains of definition can be extended
k=1
to the whole complex plane (R = ∞). It is readily verified that the equation
s
k−1 k
1 − p (1 − p)x = 0
k=1
has a unique root z 0 on the interval (1, ∞). Hence condition C1 is satisfied. The
verification of the technical conditions C2 and C3 is omitted and is left to the
interested reader. The unique root z 0 of the above equation must be numerically