Asymptotic mixed normality of maximum likelihood estimator for Ewens--Pitman partition

This paper investigates the asymptotic properties of parameter estimation for the Ewens--Pitman partition with parameters $0<α<1$ and $θ>-α$. Especially, we show that the maximum likelihood estimator (MLE) of $α$ is $n^{α/2}$-consistent and converges to a variance mixture of normal distributions, where the variance is governed by the Mittag-Leffler distribution. Moreover, we show that a proper normalization involving a random statistic eliminates the randomness in the variance. Building on this result, we construct an approximate confidence interval for $α$. Our proof relies on a stable martingale central limit theorem, which is of independent interest.