Consider sequences {Xi}mi=1 and
{Yj}nj=1 of independent random variables,
taking values in a finite alphabet, and assume that the variables X1, X2, …
and Y1, Y2, … follow the
distributions μ and v, respectively. Two variables Xi and
Yj are said to match if Xi = Yj.
Let the number of matching subsequences of length k between the two sequences, when r,
0 [les ] 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 E[W] stays bounded away from infinity, and the case
where E[W] → ∞ as m, n → ∞,
have to be treated separately. Under the assumption that
ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which
k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.
Compound Poisson approximation for extremes of moving minima in arrays of independent random variables