Sequence Comparison.pdf

126 7 Local Alignment Statistics
Because the mean step size is negative, the distribution function
is finite. Moreover, it satisfies the renewal equation
S(y)
for any 0 ≤ y < ∞. Define
S(y)=Pr[max
k≥0 S k ≤ y] (7.19)
= S(−1)+(S(y) − S(−1))
= ( 1 − Pr[t + < ∞] ) +
V (y) satisfies the renewal equation
y
∑
k=0
V (y)= ( Pr[t + < ∞] − Pr[Z + ≤ y] ) e λy +
Pr[Z + = k]S(y − k) (7.20)
V (y)=(1 − S(y))e λy . (7.21)
y
∑
k=0
e λk Pr[Z + = k]V (y − k). (7.22)
By equation (7.18), e λk Pr[Z + = k] is a probability distribution. Using the Renewal
Theorem in Section B.8, we have that
lim V (y)
y→∞
= ∑∞ y=0 eλy (Pr[t + < ∞] − Pr[Z + ≤ y])
E ( Z + e λZ+ ; t + < ∞ ) .
By (7.18), multiplying the numerator by e λ − 1 yields 1 − Pr[t + < ∞] because the
step sizes that have nonzero probability have 1 as their greatest common divisor.
Hence,
lim V (y)
y→∞
1 − Pr[t + < ∞]
= (
e λ − 1 ) E ( Z + e λZ+ ; t + < ∞ ).
Recall that Q 1 = max 0≤k