An Erdös-Rényi law with shifts

Abstract

Motivated by the comparison of DNA sequences, a generalization is given of the result of Erdös and Rényi on the length R n of the longest run of heads in the first n tosses of a coin. Consider two sequences, X 1 X 2 … X n and Y 1 Y 2 … Y n . The length of the longest matching consecutive subsequence, allowing shifts, is M n ≡ max{ m: X i + k = Y j + k for k = 1 to m, for some 0 ⩽ i, j ⩽ n − m}. Suppose that all the “letters” are independent and identically distributed. The length of the longest match without shifts has the same distribution as R n , the length of the longest head run for a biased coin with p = P( X i = Y i ), described by the Erdös-Rényi law: P( lim n → ∞ R n log 1 p (n) = 1) = 1 . For matching with shifts, our result is: P( lim n → ∞ M n log 1 p (n) = 2) = 1 . Loosely speaking, allowing shifts doubles the length of the longest match. The case of Markov chains is also handled.