A remark on a weak convergence of a linear decomposable statistics

We show that the interpretation of statistics
$$ \sum_{i=1}^N\nu_{in}l_i $$
(where $\nu_{in}$ is a frequency of the $i$-th outcome in the polynomial scheme with $N$ outcomes and $n$ trials and $l_i$ are coefficients) as a stochastic integral of the empirical process permits to investigate its asymptotic properties in a more general situation (and in a simpler manner) than in [2].