The assumption X2 bounded a.s. in Theorem 20.4 may be dropped if we slightly modify the estimate. Define ˜mn by                         m˜ n(·) = arg min f∈Wk(Rd)  1 n n i=1 |f(Xi) − Yi| 2 · I[−...

The assumption X2 bounded a.s. in Theorem 20.4 may be dropped if we slightly modify the estimate. Define ˜mn by

                        m˜ n(·) = arg min f∈Wk(Rd)  1 n n i=1 |f(Xi) − Yi| 2 · I[− log(n),log(n)]d (Xi) + λnJ2 k (f)

and set mn(x) = Tlog(n)m˜ n(x) · I[− log(n),log(n)]d (x). Show that mn is strongly consistent for all distributions of (X, Y ) with EY 2 <>d and suitable modifications of (20.26)–(20.27) hold. Hint: Use

    |mn(x) − m(x)| 2 µ(dx) =

                                     +E{|mn(X) − Y | 2 · I[− log(n),log(n)]d (X)|Dn}

                                      −E{|m(X) − Y | 2 · I[− log(n),log(n)]d (X)}.