## The Annals of Probability

- Ann. Probab.
- Volume 22, Number 4 (1994), 1993-2021.

### Critical Phenomena for Sequence Matching with Scoring

Amir Dembo, Samuel Karlin, and Ofer Zeitouni

#### Abstract

Consider two independent sequences $X_1,\ldots, X_n$ and $Y_1,\ldots, Y_n$. Suppose that $X_1,\ldots, X_n$ are i.i.d. $\mu X$ and $Y_1,\ldots, Y_n$ are i.i.d. $\mu_Y$, where $\mu_X$ and $\mu_Y$ are distributions on finite alphabets $\sum_X$ and $\sum_Y$, respectively. A score $F: \sum_X \times \sum_Y \rightarrow \mathbb{R}$ is assigned to each pair $(X_i, Y_j)$ and the maximal nonaligned segment score is $M_n = \max_{0\leq i, j \leq n - \Delta, \Delta \geq 0}\{\sum^\Delta_{l=1}F(X_{i+l}, Y_{j+l})\}$. Our result is that $M_n/\log n \rightarrow \gamma^\ast(\mu_X, \mu_Y)$ a.s. with $\gamma^\ast$ determined by a tractable variational formula. Moreover, the pair empirical measure of $(X_{i+l}, Y_{j+l})$ during the segment where $M_n$ is achieved converges to a probability measure $\nu^\ast$, which is accessible by the same formula. These results generalize to $X_i, Y_j$ taking values in any Polish space, to intrasequence scores under shifts, to long quality segments and to more than two sequences.

#### Article information

**Source**

Ann. Probab., Volume 22, Number 4 (1994), 1993-2021.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988492

**Digital Object Identifier**

doi:10.1214/aop/1176988492

**Mathematical Reviews number (MathSciNet)**

MR1331213

**Zentralblatt MATH identifier**

0834.60031

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 60F15: Strong theorems

**Keywords**

Large deviations strong laws sequence matching large segmental sums

#### Citation

Dembo, Amir; Karlin, Samuel; Zeitouni, Ofer. Critical Phenomena for Sequence Matching with Scoring. Ann. Probab. 22 (1994), no. 4, 1993--2021. doi:10.1214/aop/1176988492. https://projecteuclid.org/euclid.aop/1176988492