$$q$$-Moment Stein Kernels: Application to the Rate of Convergence in the Multidimensional CLT

In a recent study [Ann. Probab. 47 (4) 2172–2185, 2019], M. Fathi obtained new rates of convergence in the central limit theorem for the log-concave situations. His approach is based on analyzing Stein kernels using moment maps. In the present paper, we replace moment maps by $$q$$ -moment maps and describe construction of Stein kernels using $$q$$ -moment maps, which are convex solutions of a Monge–Ampère equation in [Nonlinear Anal. 239 (2024), 113417]. Then a bound on the Kantorovich–Wasserstein distance in the central limit theorem and in the context of probability distributions of negative powers is proved.