A Limit Theorem for Matching Random Sequences Allowing Deletions

In particular, if we restrict a(O) = g, a(1) = g + 1, ..., a(l) = g + I and b(O) = h, b(1) = h + 1, ..., b(l) = h + 1, the corresponding score is called the nonalignment score or the score without deletions. The score function s(x, y) for aligned pairs is 1 if x = y and ,u if x # y. In words, each match is rewarded by 1, each mismatch is penalized by ,u and each deletion by 8. Let Sn=S(Ai,...,An,Bl,...,Bn). That is, I=Al,...,An and J=Bl,...,Bn. By a standard subadditive argument (see [2]), it is easy to see that, for