Proof of theorem 1. Denoting , we have the expansion with , , and

Note that, under *H*_{ ∘ }, . Hence, assumptions **A1**, **A2**, and **A4** together with lemma 4 in Zhao & Wu (2001) entail that , whereas lemma 6 in Zhao & Wu (2001) leads us to with *Z* ∼ *N*(0,1). As in the proof of lemma 4 in Zhao & Wu (2001), lemma 1(i) in Zhao & Wu (2001) together with assumptions **A1**, **A2**, and **A4** imply that

Therefore, it only remains to be shown that

- (8)

- (9)

Using a Taylor expansion of order 1 and the fact that, under *H*_{ ∘ }, , we obtain

where *ξ*_{n} is an intermediate point between and *β*_{ ∘ }. Then, lemma 1 in Zhao & Wu (2001), the dominated convergence theorem, and the fact that from **A5**(a) has a root-*n* order of convergence entail that , concluding the proof of ‘(8)’.

Likewise, under *H*_{ ∘ }, a second-order Taylor expansion gives , where

Using lemma 1 from Zhao & Wu (2001) and assumptions **A1** and **A3**, we obtain . Therefore, as , we have , which together with the fact that implies that the first term of *nh*^{d / 2}*T*_{ n4} converges to 0 in probability. Straightforward calculations allow us to show that the second term of *T*_{n4} is negligible compared with the first one, as from **A5**(a), , concluding the proof of ‘(9)’.□

Proof of theorem 2. We have the expansion , where *T*_{nj} are defined as in the proof of theorem 1, with . Note that *T*_{n4} = *T*_{n41} + 2*T*_{n42}, where

and

As in the proof of theorem 1, we have . Besides, , where

Using lemma 1 in Zhao & Wu (2001) and the dominated convergence theorem, we can easily derive that , as **A1** and **A2** hold. Therefore, . Arguing as in theorem 1, we obtain , which entails that .

On the other hand, **A1** and **A2** and lemma 1 in Zhao & Wu (2001) imply that . Similarly, using from **A5**(b) and the Cauchy–Schwartz inequality, it is easy to see that . Besides, as in theorem 1, , and so .

Then, as in the proof of theorem 1, we obtain

which concludes the proof.□

Proof of theorem 3. Let , and note that

Using arguments analogous to those considered in the proof of theorem 1, when deriving the asymptotic behaviour of *T*_{n3} and *T*_{n4} and using **A6**, we obtain . Then, it is enough to obtain the asymptotic behaviour of . Let

where and denotes expectation with respect to the bootstrap distribution. Then, . The term is a quadratic form with . Then, as in lemma 6 in Zhao & Wu (2001), from **A1** and **A2**, the fact that is a consistent estimator of *β*_{1}, and the dominated convergence theorem, we obtain . Arguing as in lemma 6 in Zhao & Wu (2001), but conditional on **x**_{1}, … ,**x**_{n}, we can see that the conditions of theorem 1 in Hall (1984) are satisfied. Finally, arguments analogous to those considered in lemma 4 in Zhao & Wu (2001) allow us to show that **A1**, **A2**, and **A4** imply that .□