On Compound Poisson Approximation For Sequence Matching

Consider the sequences fX i g m i=1 and fY i g n j=1 of independent random variables , which take values in a nite alphabet, and assume that the variables X 1 ; X 2 ; : : : and Y 1 ; Y 2 ; : : : follow the distributions and , respectively. Two variables X i and Y j are said to match if X i = Y j. Let the number of matching subsequences of length k between the two sequences, when r, 0 r < k, mismatches are allowed, be denoted by W. In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where EW] stays bounded away from innnity, and the case where EW] ! 1 as m; n ! 1, have to be treated separately. Under the assumption that ln n= ln(mn) ! 2 (0; 1), we give conditions on the rate at which k ! 1, and on the distributions and , for which the variation distance tends to zero.