Asymptotic normality of one class of statistics in a multinomial scheme

There are $N$ cells into which $n_j$ particles of the $j$-th type are thrown independently of each other, $j=1,\dots,s$. Particles of the $j$-th type are distributed in cells with the probabilities $p_{1j},\dots,p_{Nj}$. Let
$$ L_r=\sum_{m=1}^N f_{mr}^{(N)}(\nu_{m1},\dots,\nu_{ms}), $$
where $\nu_{mj}$ is the number of particles of the $j$-th type in the $m$-th cell and $f_{mr}^{(N)}(x_1,\dots,x_s)$ are some given functions. The central limit theorem for the multidimensional random variables $(L_1,\dots,L_k)$, as $N,n_1,\dots,n_s\to\infty$, is proved.