Lower bounds for moments of global scores of pairwise Markov chains

TL;DR

This paper establishes lower bounds for the moments of global similarity scores, like the longest common subsequence, of Markov chain sequences, providing a general condition for their order of growth.

Contribution

It introduces a broad condition ensuring the moments of similarity scores grow at a rate proportional to n^{r/2} for Markov chain models.

Findings

Moments of similarity scores grow as n^{r/2} under certain conditions.

The main result applies to various Markov models including hidden Markov models.

Simulations support the theoretical condition's validity.

Abstract

Let $X_1,X_2,\ldots$ and $Y_1,Y_2,\ldots$ be two random sequences so that every random variable takes values in a finite set $\mathbb{A}$. We consider a global similarity score $L_n:=L(X_1,\ldots,X_n;Y_1,\ldots,Y_n)$ that measures the homology (relatedness) of words $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$. A typical example of such score is the length of the longest common subsequence. We study the order of central absolute moment $E|L_n-EL_n|^r$ in the case where two-dimensional process $(X_1,Y_1),(X_2,Y_2),\ldots$ is a Markov chain on $\mathbb{A}\times \mathbb{A}$. This is a very general model involving independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a general condition that guarantees that $E|L_n-EL_n|^r\asymp n^{r\over 2}$. We also perform simulations indicating the validity of the condition.