The compound Poisson distribution of the number of matches of values of a~discrete function of $s$-tuples in segments of a~sequence of random variables

A. M. Shoitov

Diskretnaya Matematika, Tome 19 (2007) no. 2, pp. 6-26

Résumé

For a sequence $\mathbf X=\{X_1,\dots,X_n,\dots\}$ of independent identically distributed random variables, we construct the $s$-tuples $Y_i(s)=(X_i,\dots,X_{i+s-1})$, $i=1,2,\dots,n$, and consider the random variables $\mathbf F_i=f(Y_i(s))$, $i=1,2,\dots$, where $f$ is a function defined on the set $\mathbf R^s$ and taking non-negative integer values. We consider the sequence $\mathbf F=\{\mathbf F_1,\mathbf F_2,\dots\}$ and study two random variables, the variable $$ \mathbf Z_n(\mathbf F)=\sum_{1\le i_1\le n}\mathbf I\{\mathbf F_{i_1}=\mathbf F_{i_2}\} $$ equal to the number of matches of symbols on a segment of length $n$ of the sequence $\mathbf F$ (here $\mathbf I\{\cdot\}$ stands for the indicator of a random event), and the variable $$ \mathbf Z'_n(\mathbf F)=\sum_{1\le i_1+s\le i_2\le n}\mathbf I\{\mathbf F_{i_1}=\mathbf F_{i_2}\} $$ equal to the number of matches of values of the function $f$ of non-overlapping $s$-tuples of a segment of the sequence $\mathbf X$ of length $n+s-1$. With the use of the Stein method, we find sufficient conditions for the distribution of the random variables $\mathbf Z_n(\mathbf F)$ and $\mathbf Z'_n(\mathbf F)$ to converge to the compound Poisson law for the function $f$ of a general form. As corollaries to these results we obtain both known and new limit theorems for the number of matches of values of a function of segments of sequences in a polynomial scheme for a series of particular types of the function $f$.