Two remarks on the multidimensional $\chi^2$ statistic

We consider a sequence of polynomial trials with $N$ outcomes and construct the multivariate statistic $\chi^2$ with the use of samples of growing sizes $n_1,\ldots,n_r$, $1\le n_1<\ldots<n_r$, $r\ge 2$, such that each subsequent sample contains the previous one. We assume that $N$ is fixed, $n_1\to\infty$, and $n_i/n_{i+1}\to\rho_i^2$, $0<\rho_i<1$, $i=1,\ldots,r-1$.
For fixed (not close) alternatives to a simple hypothesis tested, we establish the weak convergence of the distribution of the vector statistic $\chi^2$, whose components are appropriately centered and normalized, to multivariate normal and chi-square laws. In the case of convergence to the normal law, the components of the limiting normal random vector form a non-homogeneous Markov chain; the densities of transition probabilities of this chain are found.
This research was supported by the Russian Fiundation for Basic Research, grants 96–01–00531, 96–15–96092.