Double asymptotics for the chi-square statistic

Abstract

We consider distributional limit of the Pearson chi-square statistic when the number of classes m increases with the sample size n in such way that $n/\sqrt{m}\to \lambda$. Under mild moment conditions, the limit is Gaussian for $\lambda =\infty$, Poisson for finite $\lambda >0$, and degenerate for $\lambda =0$.