## Bernoulli

- Bernoulli
- Volume 24, Number 2 (2018), 971-992.

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

Raphael Hauser, Heinrich Matzinger, and Ionel Popescu

#### 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).

#### Article information

**Source**

Bernoulli, Volume 24, Number 2 (2018), 971-992.

**Dates**

Received: September 2014

Revised: November 2015

First available in Project Euclid: 21 September 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1505980885

**Digital Object Identifier**

doi:10.3150/16-BEJ823

**Mathematical Reviews number (MathSciNet)**

MR3706783

**Zentralblatt MATH identifier**

06778354

**Keywords**

convex geometry large deviations percolation theory sequence alignment

#### Citation

Hauser, Raphael; Matzinger, Heinrich; Popescu, Ionel. An upper bound on the convergence rate of a second functional in optimal sequence alignment. Bernoulli 24 (2018), no. 2, 971--992. doi:10.3150/16-BEJ823. https://projecteuclid.org/euclid.bj/1505980885