On Asymptotic Properties of Some Statistics Similar to $\chi ^2 $
Iosif Ilyich Gikhman · Theory of Probability and Its Applications · 1956
A sequence of sequences of tests is considered (independent in each sequences) where possibleoutcomes $E_1 ,E_2 , \cdots ,E_n $ have probabilities of $p_1 ,p_2 , \cdots ,p_n $ respectively, where $p_i > 0$ and $\sum _i p_i = 1$. A group of possible outcomes $(E_1 ,E_2 , \cdots ,E_n )$ is distinguished for which \[ \mathop {\lim }\limits_{N \to \infty } \mathop {\max }\limits_{1 \leqq k \leqq m} p_{i_k } = 0,{\text{ and }}p_{i_k } = \alpha _0 , \] 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.