Averaging principles and central limit theorems for multiscale McKean-Vlasov stochastic systems

In this paper, we study a class of multiscale McKean-Vlasov stochastic systems where the entire system depends on the distribution of the fast component. First of all, by the Poisson equation method we prove that the slow component converges to the solution of the averaging equation in the $L^p$ ($p\ge 2$) space with the optimal convergence rate 1/2. Then a central limit theorem is established by tightness.