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July, 1989 The Erdos-Renyi Strong Law for Pattern Matching with a Given Proportion of Mismatches
R. Arratia, M. S. Waterman
Ann. Probab. 17(3): 1152-1169 (July, 1989). DOI: 10.1214/aop/1176991262


Consider two random sequences $X_1 \cdots X_n$ and $Y_1 \cdots Y_n$ of i.i.d. letters in which the probability that two distinct letters match is $p > 0$. For each value $a$ between $p$ and 1, the length of the longest contiguous matching between the two sequences, requiring only a proportion $a$ of corresponding letters to match, satisfies a strong law analogous to the Erdos-Renyi law for coin tossing. The same law applies to matching between two nonoverlapping regions within a single sequence $X_1 \cdots X_n$, and a strong law with a smaller constant applies to matching between two overlapping regions within that single sequence. The method here also works to obtain the strong law for matching between multidimensional arrays, between two Markov chains and for the situation in which a given proportion of mismatches is required.


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R. Arratia. M. S. Waterman. "The Erdos-Renyi Strong Law for Pattern Matching with a Given Proportion of Mismatches." Ann. Probab. 17 (3) 1152 - 1169, July, 1989.


Published: July, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0688.62019
MathSciNet: MR1009450
Digital Object Identifier: 10.1214/aop/1176991262

Primary: 62E20
Secondary: 62P10

Keywords: DNA sequences , Hamming distance , Ising model , large deviations , Matching , Potts model , protein sequences

Rights: Copyright © 1989 Institute of Mathematical Statistics

Vol.17 • No. 3 • July, 1989
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