Sequence Comparison.pdf

124 7 Local Alignment Statistics
Let X i denote the score of the aligned pair at the ith position. Then, X i s are iid
random variables. Let X be a random variable with the same distribution as X i s. The
moment generating function of X is
We consider the accumulative scores:
(
E e θX) c
= ∑ p j e jθ .
j=−d
S 0 = 0,
S j =
j
∑
i=1
X i , j = 1,2,...,
and partition the walk into non-negative excursions between the successive descending
ladder points in the path:
K 0 = 0,
K i = min { }
k | k ≥ K i−1 + 1, S k < S Ki−1 , i = 1,2,.... (7.8)
Because the mean step size is negative, the K i − K i−1 are positive integer-valued
iid random variables. Define Q i to be the maximal score attained during the ith
excursion between K i−1 and K i , i.e.,
( )
Q i = max Sk − S Ki−1 , i = 1,2,.... (7.9)
K i−1 ≤k 0, S i ≤ 0,i = 1,2,...,k − 1}. (7.10)
and set t + = ∞ if S i ≤ 0 for all i. Then t + is the stopping time of the first positive
accumulative score. We define
and
Z + = S t +, t + < ∞ (7.11)
L(y)=Pr[0 < Z + ≤ y]. (7.12)
Lemma 7.1. With notations defined above,
E(K 1 )=exp
{ ∞∑
k=1
}
1
k Pr[S k ≥ 0]
(7.13)
and