ABSTRACT
The statement of this theoreirr is a little less explicit than that of Theorern 3.3.1, in particular bccanse L* # L needs to be calculat,ecl. It is actually possible to prove 1.oug11 bounds for the quantities Ek(~f71,,,,) and IE,(T(,,,~)), 2 E A*, and for the difference between L' and I,? in terms of the elements of the transition matrix II. However, id1 these quaritities can be computed simply by solving srna.11 sets of lirlcar equations if IAl is small, as is the case for the four let,tcr DNA alphabet or the t,wenty letter arnino acid alphabet, and this is the procedure that wc adopt in the riurrlerical comparisons. The bound in Theorem 3.3.2 is asymptotically better than that of Theorem 3.3.1, improving the order O($ log n) to C l ( $ ) . However, for moderate 12, the logaritlirnic factor may riot dominate the contributions from the constant elerrierits in the b o u ~ ~ d s , so that nurrrerical comparison for such 11 is also of interest.
