Show that under the assumptions of Theorem 11.5 one has for arbitrary δ > 0,                                E  |mn(x) − m(x)| 2 µ(dx)                               ≤...

Show that under the assumptions of Theorem 11.5 one has for arbitrary δ > 0,

                               E

                              ≤ cδ,L (1 + log(n))VF+ n n + (1 + δ) inf f∈Fn  |f(x) − m(x)| 2 µ(dx)

for some constant cδ,L depending only on δ and L. How does cδ,L depend on δ? Hint: Use the error decomposition

                                   |mn(x) − m(x)| 2 µ(dx)

                                   =  E{|mn(X) − Y | 2 |Dn} − E{|m(X) − Y | 2 }

                                  −(1 + δ) ·  1 n n i=1 |mn(Xi) − Yi| 2 − 1 n n i=1 |m(Xi) − Yi| 2

                            +(1 + δ) ·  1 n n i=1 |mn(Xi) − Yi| 2 − 1 n n i=1 |m(Xi) − Yi| 2 = T1,n + T2,n.