̂ ℧ = n − 1 ∑ n i $_{=1}$ϵ$_{i}$ ( ̂ h ( X$_{i}$ ) − h$_{0}$ ( X$_{i}$ )) ˜ b K $_{w}$( X$_{i}$ ) ˜ b K $_{w}$( X$_{i}$ ) $^{′}$. By the inequality 2 | a | ≤ 1 + a $^{2}$, it follows that n ‖ ̂ $^{℧}$‖ ≤ ‖ ̂ h − h$_{0}$ ‖ ∞ ,w × ∥ n − 1 n ∑ i =1 (1 + ϵ 2 $_{i}$) ˜ b K $_{w}$( X$_{i}$ ) ˜ b K $_{w}$( X$_{i}$ ) ′ ∥ (116) ′ = ‖ h − h$_{0}$ ‖ ∞ ,w × ∥ ˜ B ′ w ˜ B$_{w}$/n + ̂ Ω ∥ = o$_{p}$ (1) × O$_{p}$ (1) = o$_{p}$ (1) (117) because ‖ ̂ h − h$_{0}$ ‖ ∞ ,w = o$_{p}$ (1), ‖ ˜ B ′ w ˜ B$_{w}$/n ‖ = O$_{p}$ (1) by Assumption 5, and ‖ ̂ Ω ‖ = O$_{p}$ (1) by Assumption 8. Proof of Theorem 3.2. This follows from Lemma 3.2. First, Assumption 5 is satisfied for i.i.d. and β -mixing data under the respective conditions on K (see Lemmas 2.1 and 2.2). Moreover, Lemma 3.1 shows that Assumption 8 and the condition ‖ ˜ B ′ w ˜ B$_{w}$/n − I$_{K}$ ‖ = o$_{p}$ ( K − 1 / $^{2}$) is satisfied for weakly dependent data under the respective conditions on K (see Lemma 2.2). Therefore Theorem 3.1 may be applied for asymptotic normality of f ( ̂ h ). To apply Lemma 3.2 it remains to show that h h$_{0}$ ∞ ,w = o$_{p}$ (1). But h h$_{0}$ ∞ = o$_{p}$ (1) by To apply Lemma 3.2 it remains to show that ‖ ̂ h − h$_{0}$ ‖ ∞ ,w = o$_{p}$ (1). But ‖ ˜ h − h$_{0}$ ‖ ∞ = o$_{p}$ (1) by assumption, and ‖ ̂ h − ˜ h ‖ ∞ = O$_{p}$ ( ζ$_{K,n}$λ$_{K,n}$ √ (log n ) /n ) = o$_{p}$ (1) by Lemmas 2.3, 2.1 and 2.2 under the conditions on K . Proof of Corollary 4.1. Follows from Theorem 4.1 with t = Cσ$_{n}$ √ log( d$_{1}$ + d$_{2}$ ) for sufficiently large C , and applying the condition R$_{n}$ √ log( d$_{1}$ + d$_{2}$ ) = o ( σ$_{n}$ ). Proof of Theorem 4.2. By Berbee’s lemma (enlarging the probability space as necessary) the process { X$_{i}$ } can be coupled with a process X ∗ i such that Y$_{k}$ := { X$_{(}$$_{k}$$_{−}$$_{1)}$$_{q}$$_{+1}$, . . . , X$_{kq}$ } and Y ∗ k := { X ∗ ( k − 1) q $_{+1}$, . . . , X ∗ kq } are identically distributed for each k ≥ 1, $^{P}$( Y$_{k}$ = Y ∗ k ) ≤ β ( q ) for each k ≥ 1 and { Y ∗ 1 , Y ∗ 3 , . . . } are independent and { Y ∗ 2 , Y ∗ 4 , . . . } are independent (see Lemma 2.1 of Berbee (1987)). Let I$_{e}$ and I$_{o}$ denote the indices of { 1 , . . . , n } corresponding to the odd- and even-numbered blocks, and I$_{r}$ the indices in the remainder, so I$_{r}$ = q [ n/q ] + 1 , . . . , n when q [ n/q ] < n and I$_{r}$ =  when q [ n/q ] = n . 45 7.3 Proofs for Section 4