Let M ∈ N and let GM be the set of all (multivariate) polynomials of degree M (or less, in each coordinate). Let Pn = Pn(Dn) be a data-dependent partition and set                              GM ◦ Pn...

Let M ∈ N and let GM be the set of all (multivariate) polynomials of degree M (or less, in each coordinate). Let Pn = Pn(Dn) be a data-dependent partition and set

                             GM ◦ Pn =  f : Rd → R : f = A∈Pn gAIA for some gA ∈ GM (A ∈ Pn) .

Define the estimate mn by (13.2) and (13.3) with Gc replaced by GM. Show that (13.6)–(13.10) imply

                             |mn(x) − m(x)| 2 µ(dx) → 0 (n → ∞) a.s