Poisson approximation for the number of long match patterns in random sequences

Let $X_1 , \ldots ,X_m ,Y_1 , \ldots ,Y_n $ be independent identically distributed random variables with discrete state space. We estimate the rate of convergence in the limit theorems for the number of long match patterns and for the length of the longest match pattern in random sequences $X_1 , \ldots ,X_m ,Y_1 , \ldots ,Y_n $. The results improve the corresponding ones received by Zubkov–Mikhailov, Arratia–Gordon–Waterman, and others.