Variational convergence and kernel density estimation by PDE

Title: Variational convergence and kernel density estimation by PDE

This study concerns the variational convergence and their applications to kernel density estimation. Uniform convergence is not suitable for an objective function with differential operator. However, it is still possible to prove that, for any reasonable choice of the weaker topology, the minimum points and the minimum values of the empirical functionals converge to the true minimum point and to the minimum value of true functional. In this talk, as a reasonable choice of the topology we choose the Mosco-convergence, that is the "weakest" notion of convergence for sequences of convex functional which allows to approach the limit on corresponding minimization problems. On this way, a limit problem of a kernel density estimate by partial differential equation method is analyzed.