On Asymptotic Properties of Some Statistics Similar to $\chi^2$

A sequence of sequences of tests is considered (independent in each sequences) where possibleoutcomes $E_1,E_2,\dots,E_n$ have probabilities of $p_1,p_2,\dots,p_n$ respectively, where $p_i>0$ and $\sum_i p_i=1$. A group of possible outcomes $(E_1,E_2,\dots,E_n)$ is distinguished for which
$$\lim_{N\to\infty}\max_{1\leq k\leq m}p_{i_k}=0,\text{ и }\sum_{k=1}^m p_{i_k}=\alpha_0,$$
where $m$ and $\alpha_0$ are independent of the number of sequences $N$.
Theorems are given for sequences of sequences of certain statistics similar in structure to $\chi^2$, which show that these sequences converge to appropriate continuous Markov processes.