Estimate of the Accuracy of the Compound Poisson Approximation for the Distribution of the Number of Matching Patterns

Let $X_1,\dots,X_m$ and $Y_1,\dots,Y_n$ be two sequences of independent identically distributed random variables taking on values $1,2,\dots$ . By means of a particular version of the Stein method we construct an estimate of the accuracy of approximation for the distribution of the number of matching patterns of outcomes $X_i,\dots,X_{i+s-1}$ of a given length $s$ in the first sequence with the patterns of outcomes $Y_j,\dots,Y_{j+s-1}$ in the second sequence. The approximating distribution is the distribution of the sum of Poisson number of independent random variables with geometric distribution.