Lower bounds for moments of global scores of pairwise Markov chains

Let X1,… and Y1,… be random sequences taking values in a finite set A. We consider a similarity score Ln≔L(X1,…,Xn;Y1,…,Yn) that measures the homology of words (X1,…,Xn) and (Y1,…,Yn). A typical example is the length of the longest common subsequence. We study the order of moment E|Ln−ELn|r in the case where the two-dimensional process (X1,Y1),(X2,Y2),… is a Markov chain on A×A. This general model involves independent Markov chains, hidden Markov models, Markov switching models and many more. Our main result establishes a condition that guarantees that E|Ln−ELn|r≍nr2. We also perform simulations indicating the validity of the condition.