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 . Specifically, we show that the maximum-likelihood estimator (MLE) of α is nα/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.