A moving chi-square (Q2708300)

The authors consider the following (more general) problem: Let \((x_t)\) be a sequence of i.i.d. (independent, identically distributed) random variables taking values from the set \(\{1,2,\dots, N\}\). The samples used for the construction of \(\chi^2\)-statistics are obtained from partitioning \((x_t)\) into disjoint intervals of length \(n\), and \(s\)-tuples of size \(ns\) for a number \(t\): \((x(t, n),x(t+ 1,n),\dots, x(t+ s-1, n))\). Such \(s\)-tuple samples of size \(ns\), for the numbers \(1\leq t_1< t_2<\cdots< t_r\), are then utilized to construct the joint \(r\)-dimensional distribution of \(\chi^2\)-statistics. The paper shows that the joint \(r\)-dimensional vector distribution of \(\chi^2\)-statistics converges to a certain limit distribution, with \(n\to\infty\) and \(N\), \(r\) being fixed. For the obtained limit distribution, its expression of the Laplace transform is computed, and its Gaussian approximation is provided.