Let p = k + β for some k ∈ N0 and β ∈ (0, 1]. Show that for a suitable choice of the parameters K = Kn and M the estimate in Problem 19.1 satisfies                         E  |mn,(Kn,M)(x) − m(x)| 2...

Let p = k + β for some k ∈ N0 and β ∈ (0, 1]. Show that for a suitable choice of the parameters K = Kn and M the estimate in Problem 19.1 satisfies

                        E  |mn,(Kn,M)(x) − m(x)| 2 µ(dx) ≤ c  C 2d 2p+d n− 2p 2p+d

for every distribution of (X, Y ) with X ∈ [0, 1]d a.s., |Y | ≤ L a.s., and m (p, C)- smooth.