Asymptotic Normality and Convergence Rates for Tsallis Entropy Estimators via Stabilization Techniques

We study nearest-neighbor-based estimators of Tsallis entropy associated with Poisson and binomial point processes on general metric measure spaces. In this study, by combining existing stabilization methods with the validation of the estimator's local k-nearest-neighbor structure, we investigate nearest-neighbor-based Tsallis entropy estimators under Poisson and binomial distributed input data. Rather than proposing a new second-order Poincaré inequality, this paper details and clearly presents stabilization-based normal approximation bounds for Tsallis-type k-NN functionals. We establish asymptotic normality and derive explicit convergence rates for the Kolmogorov distance. Our analysis avoids explicit score-function decompositions and instead relies on flexible localizations of add-one costs, which simplify the treatment of higher-order terms. Under natural stabilization and moment conditions, the resulting bounds recover the classical normal approximation rates s-1/2 and n-1/2 and extend corresponding results for Shannon and Rényi entropy estimators. We further illustrate the scope of the framework through examples involving Tsallis entropy functionals, weighted k-NN Shannon entropy estimators. The examples provided highlight the benefits of stabilization-based normal approximations for non-parametric statistical inference in complex spatial and high-dimensional settings.