Asymptotic minimaxity of chi-square tests

We consider the asymptotic behavior of chi-square tests when a number $k_n$ of cells increases as the sample size $n$ grows. For such a setting we show that a sequence of chi-square tests is asymptotically minimax if $k_n = o(n^2)$ as $n \to \infty$. The proof makes use of a theorem about asymptotic normality of chi-square test statistics obtained under new assumptions.