Matchings of values of a function of segments of a sequence of independent trials (Q2708273)

Let \(X_1,X_2,\dots\) be a sequence of independent and identically distributed random variables. For an arbitrary natural number \(n\) define the sequence of \((n-1)\)-dependent vectors \(Y_1,Y_2,\dots\) by \(Y_i= (X_i,\dots, X_{i+n-1})\). Furthermore, consider a Borel function \(\overline f(\overline a)= (f_1(\overline a),\dots, f_m(\overline a))\), \(\overline a\in\mathbb{R}^n\), that assumes its values in the \(m\)th Cartesian power \(\{0,1,\dots\}^m\). Set \(F_i(c)= f_c(Y_i)\), \(c= 1,\dots, m\), and for any integer \(k\geq 2\) and arbitrary integers \(1\leq i_1< i_2<\cdots< i_k\leq N\) define the random variables \(\eta(i_1,\dots, i_k;c;n)= 1\) if \(F_{i_1}(c)= F_{i_k}(c)\neq 0\) and \(\eta(i_1,\dots, i_k; c;n)= 0\) otherwise. Moreover, let \(J= \{(i_1,\dots, i_k):1\leq i_1<\cdots< i_k\leq N\}\). The random variables \(\xi_c(N)= \sum_{(i_1,\dots, i_k)\in J}\eta(i_1,\dots, i_k; c;n)\), \(c=1,\dots, m\), are then interpreted as counts of the \(k\)-fold matchings of non-zero symbols on a segment of length \(N\) of the sequence \(F(1),\dots, F(m)\). Under certain assumptions on functions \(\overline f\), the author establishes weak convergences to the multivariate normal law and to the chi-square law as \(N\to\infty\). Applications to the problem of \(k\)-fold imperfect matchings of \(n\)-tuples in a sequence of polynomial trials are also given.