An upper bound on the convergence rate of a second functional in optimal sequence alignment

Abstract

Consider finite sequences $X_{[1,n]}=X_{1},\ldots,X_{n}$ and $Y_{[1,n]}=Y_{1},\ldots,Y_{n}$ of length $n$, consisting of i.i.d. samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $(\ln(n))^{1/4}n^{3/4}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun in (J. Stat. Phys. 153 (2013) 512–529).