Double asymptotics for the chi-square statistic

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 ${\lambda} = 0$.