Prove Lemma 21.1. Hint: Start with the error decomposition
|mn,(k,λ)(x) − m(x)| 2 µ(dx) = T1,n + T2,n,
where
T1,n = E F |mn,(k,λ)(X) − Y | 2 |Dn G − E(|m(X) − Y | 2 )
−2 1 n n i=1 |mn,(k,λ)(Xi) − Yi| 2 + λJ2 k ( ˜mn,(k,λ))
− 1 n n i=1 |m(Xi) − Yi| 2 + penn(k, λ)
and
T2,n = 2 1 n n i=1 |mn,(k,λ)(Xi) − Yi| 2 + λJ2 k ( ˜mn,(k,λ)) − 1 n n i=1 |m(Xi) − Yi| 2 + penn(k, λ) .
As in the proof of Theorem 21.2 show, for n sufficiently large and any t > 0,
P{T1,n > t} ≤ ∞ l=1 c3 exp −c4 n(t + 2l penn(k, λ)) L4 ≤ c3 exp −c4 n · t L4 .
Conclude
P T1,n > L4 log(n) n ≤ c3 · exp(−c4 log(n)) ≤ η
for n sufficiently large, which implies the assertion.