On a class of statistics of polynomial samples

We consider $M\ge1$ independent samples each of which is a realisation of some polynomial scheme. The number of outcomes $N$ and the number of samples $M$ are fixed, the sample sizes grow without bound. We study the asymptotic properties of statistics of the form $g(\overline\xi)$, where $g(\overline\xi)$ is a differentiable function of $MN$ real-valued variables, is the vector of relative frequencies of outcomes in samples. Statistics of such a kind are playing an important part in the applied statistical analysis.
In this research we expand the capabilities of the well-known $\delta$-method (the linearisation method) in the case of polynomial samples. We prove the asymptotic normality and convergence in distribution of the statistics $g(\overline\xi)$ to quadratic forms of normal random variables (both for the fixed probabilities of outcomes in the case of the null hypothesis and for the case of contigual alternatives to it). We give conditions for both types of convergence.