A POISSON-TYPE LIMIT THEOREM FOR THE NUMBER OF PAIRS OF MATCHING SEQUENCES

摘要

Two sequences X-1,..., X-m and Y-1,..., Y-n are considered constituted by independent identically distributed random variables within each of the sequences taking on values in the set {1, 2,...}. We study the distribution of the number N-d of such pairs of s-patterns ((X) over bar (i), (Y) over bar (j)), where (X) over bar (i) = (X-i,..., Xi+ s-1), (Y) over bar (j) = (Y-j,...,Yj+s-1), in which the s-patterns (X) over bar (i) and (Y) over bar (j) differ by a relatively small number of elements d. It is shown that if m, n, s ->infinity,d = o(s/log s), and the distributions of the elements of the sequences vary in such a way that the probability P{Xi = Yj} and ENd converge to some limiting values, then the distribution of N-d converges to a compound Poisson distribution. The value of the parameter d plays a role only to provide, passing to the limit, the needed rate of the parameters involved and has no influence on the form of the limit distribution. This limit distribution has the same form as that for the number of pairs ((X) over bar (i), (Y) over bar (j)), in which (X) over bar (i) = (Y) over bar (j).