On the power of the chi-square test with increasing number of class-intervals

The paper deals with testing a simple hypothesis against a sequence of simple alternatives converging to the hypothesis at rate $n^{-1/z}$, $n$ being the sample size.
It is known that the power of the chi-square test with $r$ equiprobable class-intervals tends to the test size if $r\to\infty$ as $n\to\infty$. Here it is shown that, in case of not equiprobable class-intervals, the power tends to a certain nondegenerate limit as $r/n\to c>0$ and, if $r/n\to\infty$, then the test behaves like a locally most powerful test against a specific sequence of alternatives depending on the behaviour of the class-intervals probabilities.