The asymptotic behavior of the Pearson statistic

Some limit theorems are proved for some functionals of the Pearson statistic constructed from the polynomial distribution with parameters $n$ and $p_k$, $k=1,2,\dots$, $s=s(n)$, under the assumption that $\inf_{n}\{n\min_{1\le k\le s}p_k\}>0$, $s\to \infty$, $n\min\{p_k: k\in W_n\}\longrightarrow \infty$, $N_n/s\to 1$ as $n\to \infty$, where $N_n$ is the number of elements in the set $W_n\subset \{1,2,\dots ,s\}$. In particular, multivariate and functional limit theorems are proved for this statistic. As a whole, the statements proved in this paper demonstrate that the Pearson statistic in many respects behaves as an asymptotically normal sum of independent random variables.