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 $n/\sqrt{m} \to λ$. Under mild moment conditions, the limit is Gaussian for $λ = \infty$, Poisson for finite $λ > 0$, and degenerate for $λ = 0$.

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