Let mn be defined as in Problem 12.2. Show that for every distribution of (X, Y ) which satisfies |X| bounded a.s., |Y | bounded a.s., and m (p, C)-smooth one has, for n sufficiently large,...

Let mn be defined as in Problem 12.2. Show that for every distribution of (X, Y ) which satisfies |X| bounded a.s., |Y | bounded a.s., and m (p, C)-smooth one has, for n sufficiently large,

|mn(x) − m(x)| 2 µ(dx) ≤ c · C 2 2p+1 · log(n) 5 n 2p 2p+1 .

May 23, 2022