Asymptotic behaviors of multiscale McKean-Vlasov stochastic systems

In this paper, we investigate a class of multiscale McKean-Vlasov stochastic systems, where the entire system depends on the distributions of both fast and slow components. First of all, by applying the Poisson equation method, we prove that the slow component converges to the solution of the averaging equation in the $L^p$ ($p\geq 2$) space with the optimal convergence order $\frac12$. Then we establish a central limit theorem for these systems and derive the weak convergence rate using the Poisson equation technique and the regularity properties of the associated Cauchy problem.