Decomposable statistics in a polynomial scheme. I

Let a random vector $\nu=(\nu_1,\nu_2,\dots,\nu_N)$ have the polynomial distribution with parameters $n$; $p_1,p_2,\dots,p_N$. A random variable $\displaystyle L(\nu)=\sum_{m=1}^N f_m(\nu_m)$, where $f_1(x),f_2(x),\dots,f_N(x)$ are arbitrary given functions, is called a decomposable statistic.
The paper deals with the limiting laws of the distribution of decomposable statistics as $n$, $N\to\infty$ for small samples, i. e. when $\displaystyle\max_m np_m\le c<\infty$, and under some weak constraints on functions $f_m(x)$. The class of decomposable statistics includes statistic $\chi^2$, likelihood ratios, linear combinations of random variables $\mu_r$, where $\mu_r$ is the number of the coordinates of $\nu$ equal to $r$, and others.