In this section, we give some lemmas, which are very useful for proving Theorem 1.
Lemma 2.1
For the model defined by (1), we have
$$ \mathbb{E} \biggl[\frac{1}{\omega (Y_{i})} \biggr]=\frac{1}{\mu },\qquad \mathbb{E} \biggl[\frac{\mu \varphi _{j,k}(Y_{i})}{\omega (Y_{i})} \biggr]= \alpha _{j,k}\quad \textit{and}\quad \mathbb{E} \biggl[\frac{\mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})} \biggr]= \beta _{j,k}. $$
Proof
This lemma can be proved by the same arguments of Kou and Guo [10]. □
Lemma 2.2
Let
\(f\in B_{r,q}^{s}\ (1\leq r,q<+\infty , s>1/r)\), and let
\(\omega (y)\)
be a nonincreasing function such that
\(\omega (y) \sim 1\). If
\(2^{j}\leq n\)
and
\(1\leq p<+\infty \), then
$$ \mathbb{E} \bigl[ \vert \widehat{\alpha }_{j_{0},k}-\alpha _{j_{0},k} \vert ^{p} \bigr]\lesssim n^{-\frac{p}{2}},\qquad \mathbb{E} \bigl[ \vert \widehat{\beta } _{j,k}-\beta _{j,k} \vert ^{p} \bigr]\lesssim n^{-\frac{p}{2}}. $$
Proof
Because the proofs of both inequalities are similar, we only prove the second one. By the definition of \(\widehat{\beta }_{j,k}\) we have
$$\begin{aligned} \vert \widehat{\beta }_{j,k}-\beta _{j,k} \vert \leq \Biggl\vert \frac{ \widehat{\mu }_{n}}{\mu } \Biggl(\frac{\mu }{n}\sum _{i=1}^{n} \frac{ \psi _{j,k}(Y_{i})}{\omega (Y_{i})}-\beta _{j,k} \Biggr) \Biggr\vert + \biggl\vert \beta _{j,k} \widehat{\mu }_{n} \biggl(\frac{1}{\mu }-\frac{1}{ \widehat{\mu }_{n}} \biggr) \biggr\vert . \end{aligned}$$
Note that the definition of \(\widehat{\mu }_{n}\) and \(\omega (y) \sim 1\) imply \(|\widehat{\mu }_{n}|\lesssim 1\). We have \(B_{r,q}^{s}( \mathbb{R})\subseteq B_{\infty , \infty }^{s-1/r}(\mathbb{R})\) in the case of \(s>\frac{1}{r}\); then \(f\in B_{\infty , \infty }^{s-1/r}( \mathbb{R})\) and \(\|f\|_{\infty }\lesssim 1\). Moreover, \(|\beta _{j,k}|=| \langle f, \psi _{j,k}\rangle |\lesssim 1\) by the Cauchy–Schwarz inequality and the orthonormality of wavelet functions. Hence, we have the following conclusion:
$$\begin{aligned} \mathbb{E} \bigl[ \vert \widehat{\beta }_{j,k}- \beta _{j,k} \vert ^{p} \bigr] \lesssim \mathbb{E} \Biggl[ \Biggl\vert \frac{1}{n}\sum_{i=1}^{n} \frac{ \mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}-\beta _{j,k} \Biggr\vert ^{p} \Biggr]+ \mathbb{E} \biggl[ \biggl\vert \frac{1}{\mu }- \frac{1}{\widehat{\mu }_{n}} \biggr\vert ^{p} \biggr]. \end{aligned}$$
(10)
Then we need to estimate \(T_{1}:=\mathbb{E} [ \vert \frac{1}{n} \sum_{i=1}^{n} \frac{\mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}- \beta _{j,k} \vert ^{p} ]\) and \(T_{2}:=\mathbb{E} [ \vert \frac{1}{ \mu }-\frac{1}{\widehat{\mu }_{n}} \vert ^{p} ]\).
• An upper bound for \(T_{1}\). Taking \(\eta _{i}:=\frac{\mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}-\beta _{j,k}\), we get
$$ T_{1}=\mathbb{E} \Biggl[ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n}\eta _{i} \Biggr\vert ^{p} \Biggr]= \biggl(\frac{1}{n} \biggr)^{p}\mathbb{E} \Biggl[ \Biggl\vert \sum _{i=1}^{n}\eta _{i} \Biggr\vert ^{p} \Biggr]. $$
Note that ψ is a function of bounded variation (see Liu and Xu [12]). We can get \(\psi :=\widetilde{\psi }-\overline{ \psi }\), where ψ̃ and ψ̅ bounded nonnegative nondecreasing functions. Define
$$ \widetilde{\eta }_{i}:=\frac{\mu \widetilde{\psi }_{j,k}(Y_{i})}{ \omega (Y_{i})}-\widetilde{\beta }_{j,k}, \qquad \overline{\eta }_{i}:=\frac{\mu \overline{\psi }_{j,k}(Y_{i})}{\omega (Y_{i})}- \overline{\beta }_{j,k} $$
with \(\widetilde{\beta }_{j,k}:=\langle f, \widetilde{\psi }_{j,k} \rangle \) and \(\overline{\beta }_{j,k}:=\langle f, \overline{\psi } _{j,k}\rangle \). Then \(\eta _{i}=\widetilde{\eta }_{i}-\overline{ \eta }_{i}\), \(\beta _{j,k}=\widetilde{\beta }_{j,k}-\overline{\beta } _{j,k}\), and
$$\begin{aligned} T_{1}= \biggl(\frac{1}{n} \biggr)^{p}\mathbb{E} \Biggl[ \Biggl\vert \sum _{i=1}^{n} (\widetilde{\eta }_{i}- \overline{\eta }_{i} ) \Biggr\vert ^{p} \Biggr]\lesssim \biggl(\frac{1}{n} \biggr)^{p} \Biggl\{ \mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n}\widetilde{ \eta }_{i} \Biggr\vert ^{p} \Biggr]+ \mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n}\overline{ \eta }_{i} \Biggr\vert ^{p} \Biggr] \Biggr\} . \end{aligned}$$
(11)
Similar arguments as in Lemma 2.1 show that \(\mathbb{E}[ \widetilde{\eta }_{i}]=0\). The function \(\frac{\widetilde{\psi }_{j,k}(y)}{ \omega (y)}\) is nondecreasing by the monotonicity of \(\widetilde{\psi }(y)\) and \(\omega (y)\). Furthermore, we get that \(\{\widetilde{\eta }_{i}, i=1, 2, \ldots , n\}\) is negatively associated by Lemma 1.1. On the other hand, it follows from (1) and \(\omega (y)\sim 1\) that
$$\begin{aligned} \mathbb{E} \bigl[ \vert \widetilde{\eta }_{i} \vert ^{p} \bigr]\lesssim \mathbb{E} \biggl[ \biggl\vert \frac{\mu \widetilde{\psi }_{j,k}(Y_{i})}{ \omega (Y_{i})} \biggr\vert ^{p} \biggr]\lesssim \int _{[0,1]} \bigl\vert \widetilde{\psi }_{j,k}(y) \bigr\vert ^{p}f(y)\,dy\lesssim 2^{j(p/2-1)}. \end{aligned}$$
(12)
In particular, \(\mathbb{E} [|\widetilde{\eta }_{i}|^{2} ] \lesssim 1\). Recall Rosenthal’s inequality [12]: if \(Y_{1}, Y_{2}, \ldots , Y_{n}\) are negatively associated random variables such that \(\mathbb{E}[Y_{i}]=0\) and \(\mathbb{E}[|Y_{i}|^{p}]< \infty \), then
$$ \mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n}Y_{i} \Biggr\vert ^{p} \Biggr] \lesssim \textstyle\begin{cases} \sum_{i=1}^{n}\mathbb{E} [ \vert Y_{i} \vert ^{p} ]+ (\sum_{i=1}^{n}\mathbb{E} [ \vert Y_{i} \vert ^{2} ] )^{{p}/{2}}, & \text{$p>2$;} \\ (\sum_{i=1}^{n}\mathbb{E} [ \vert Y_{i} \vert ^{2} ] ) ^{{p}/{2}}, & \text{$1\leq p\leq 2$.} \end{cases} $$
From this we clearly have
$$ \mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n} \widetilde{\eta }_{i} \Biggr\vert ^{p} \Biggr]\lesssim \textstyle\begin{cases} n2^{j(p/2-1)}+n^{p/2}, & \text{$p>2$;} \\ n^{p/2}, & \text{$1\leq p\leq 2$.} \end{cases} $$
This, together with \(2^{j}\leq n\), shows that \(\mathbb{E} [ \vert \sum_{i=1}^{n}\widetilde{\eta }_{i} \vert ^{p} ]\lesssim n ^{p/2}\). Similarly, \(\mathbb{E} [ \vert \sum_{i=1}^{n}\overline{ \eta }_{i} \vert ^{p} ]\lesssim n^{p/2}\). Combining these with (11), we get that
$$\begin{aligned} T_{1}\lesssim \biggl(\frac{1}{n} \biggr)^{p} \Biggl\{ \mathbb{E} \Biggl[ \Biggl\vert \sum _{i=1}^{n}\widetilde{\eta }_{i} \Biggr\vert ^{p} \Biggr]+\mathbb{E} \Biggl[ \Biggl\vert \sum _{i=1}^{n}\overline{\eta }_{i} \Biggr\vert ^{p} \Biggr] \Biggr\} \lesssim n^{-\frac{p}{2}}. \end{aligned}$$
(13)
• An upper bound for \(T_{2}\). It is easy to see from the definition of \(\widehat{\mu }_{n}\) that
$$\begin{aligned} T_{2}=\mathbb{E} \biggl[ \biggl\vert \frac{1}{\mu }-\frac{1}{\widehat{\mu } _{n}} \biggr\vert ^{p} \biggr]= \biggl(\frac{1}{n} \biggr)^{p}\mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n} \biggl( \frac{1}{\omega (Y_{i})}-\frac{1}{ \mu } \biggr) \Biggr\vert ^{p} \Biggr]. \end{aligned}$$
(14)
Defining \(\xi _{i}:=\frac{1}{\omega (Y_{i})}-\frac{1}{\mu }\), we obtain that \(\mathbb{E}[\xi _{i}]=0\) and \(\mathbb{E}[|\xi _{i}|^{p}]\lesssim 1\) by Lemma 2.1 and \(\omega (y)\sim 1\). In addition, by the monotonicity of \(\omega (y)\) and Lemma 1.1 we know that \(\xi _{1}, \xi _{2}, \ldots , \xi _{n}\) are also negatively associated. Then using Rosenthal’s inequality, we get
$$ \mathbb{E} \Biggl[ \Biggl\vert \sum_{i=1}^{n} \xi _{i} \Biggr\vert ^{p} \Biggr] \lesssim \textstyle\begin{cases} n+n^{p/2}, & \text{$p>2$;} \\ n^{p/2}, & \text{$1\leq p\leq 2$.} \end{cases} $$
Hence
$$\begin{aligned} T_{2}= \biggl(\frac{1}{n} \biggr)^{p}\mathbb{E} \Biggl[ \Biggl\vert \sum _{i=1}^{n}\xi _{i} \Biggr\vert ^{p} \Biggr] \lesssim n^{-\frac{p}{2}}. \end{aligned}$$
(15)
Finally, by (10), (13), and (15) we have
$$ \mathbb{E} \bigl[ \vert \widehat{\beta }_{j,k}-\beta _{j,k} \vert ^{p} \bigr]\lesssim n^{-\frac{p}{2}}. $$
This ends the proof. □
Lemma 2.3
Let
\(f\in B_{r,q}^{s}\)
\((1\leq r,q<+\infty , s>1/r)\)
and
\(\widehat{\beta }_{j,k}\)
be defined by (7). If
\(\omega (y)\)
is a nonincreasing function, \(\omega (y)\sim 1\), and
\(2^{j}\leq \frac{n}{ \ln n}\), then for each
\(\lambda >0\), there exists a constant
\(\kappa >1\)
such that
$$ \mathbb{P} \bigl\{ \vert \widehat{\beta }_{j,k}-\beta _{j,k} \vert \geq \kappa t_{n} \bigr\} \lesssim 2^{-\lambda j}. $$
Proof
By the same arguments of (10) we can obtain that
$$\begin{aligned} \mathbb{P} \bigl\{ \vert \widehat{\beta }_{j,k}- \beta _{j,k} \vert \geq \kappa t_{n} \bigr\} \leq{}& \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n} \sum _{i=1}^{n} \biggl(\frac{1}{\omega (Y_{i})}- \frac{1}{\mu } \biggr) \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} \\ &{}+\mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n} \biggl(\frac{ \mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}-\beta _{j,k} \biggr) \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} . \end{aligned}$$
(16)
To estimate \(\mathbb{P} \{ \vert \frac{1}{n}\sum_{i=1} ^{n} (\frac{1}{\omega (Y_{i})}-\frac{1}{\mu } ) \vert \geq \frac{\kappa t_{n}}{2} \}\), we also define \(\xi _{i}:=\frac{1}{ \omega (Y_{i})}-\frac{1}{\mu }\). Then Lemma 2.1 implies that \(\mathbb{E}[\xi _{i}]=0\). Moreover, \(|\xi _{i}|\lesssim 1\) and \(\mathbb{E}[|\xi _{i}|^{2}]\lesssim 1\) thanks to \(\omega (y)\sim 1\). On the other hand, because of the monotonicity of \(\omega (y)\) and Lemma 1.1, \(\xi _{1}, \xi _{2}, \ldots , \xi _{n}\) are also negatively associated.
Recall Bernstein’s inequality [12]: If \(Y_{1}, Y_{2}, \ldots , Y_{n}\) are negatively associated random variables such that \(\mathbb{E}[Y_{i}]=0\), \(|Y_{i}|\leq M<\infty \), and \(\mathbb{E}[|Y _{i}|^{2}]=\sigma ^{2}\), then for each \(\varepsilon >0\),
$$ \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n}Y_{i} \Biggr\vert \geq \varepsilon \Biggr\} \lesssim \exp \biggl(-\frac{n\varepsilon ^{2}}{2(\sigma ^{2}+\varepsilon M/3)} \biggr). $$
Therefore, by the previous arguments for \(\xi _{i}\) and \(t_{n}=\sqrt{\frac{ \ln n}{n}}\), we derive
$$ \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n} \biggl(\frac{1}{ \omega (Y_{i})}- \frac{1}{\mu } \biggr) \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} \lesssim \exp \biggl(-\frac{(\ln n) \kappa ^{2}/4}{2(\sigma ^{2}+\kappa /6)} \biggr). $$
Then there exists \(\kappa >1\) such that \(\exp (-\frac{(\ln n) \kappa ^{2}/4}{2(\sigma ^{2}+\kappa /6)} )\lesssim 2^{-\lambda j}\) with fixed \(\lambda >0\). Hence
$$\begin{aligned} \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n} \biggl(\frac{1}{ \omega (Y_{i})}- \frac{1}{\mu } \biggr) \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} \lesssim 2^{-\lambda j}. \end{aligned}$$
(17)
Next, we estimate \(\mathbb{P} \{ \vert \frac{1}{n}\sum_{i=1}^{n} (\frac{\mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}-\beta _{j,k} ) \vert \geq \frac{\kappa t_{n}}{2} \}\). By to the same arguments of (11) we get
$$\begin{aligned} \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n}\eta _{i} \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} \leq \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n} \sum_{i=1}^{n} \widetilde{\eta }_{i} \Biggr\vert \geq \frac{\kappa t _{n}}{4} \Biggr\} +\mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1} ^{n}\overline{\eta }_{i} \Biggr\vert \geq \frac{\kappa t_{n}}{4} \Biggr\} . \end{aligned}$$
(18)
It is easy to see from the definition of \(\widetilde{\eta }_{i}\) and Lemma 2.1 that \(\mathbb{E}[\widetilde{\eta }_{i}]=0\). Moreover, \(\mathbb{E}[|\widetilde{\eta }_{i}|^{2}]\lesssim 1\) by (12) with \(p=2\). Using \(\omega (y)\sim 1\), we get \(\vert \frac{\mu \widetilde{\psi }_{j,k}(Y_{i})}{\omega (Y_{i})} \vert \lesssim 2^{j/2}\) and \(|\widetilde{\eta }_{i}|\leq \vert \frac{\mu \widetilde{\psi } _{j,k}(Y_{i})}{\omega (Y_{i})} \vert +\mathbb{E} [ \vert \frac{ \mu \widetilde{\psi }_{j,k}(Y_{i})}{\omega (Y_{i})} \vert ] \lesssim 2^{j/2}\). Then it follows from Bernstein’s inequality, \(2^{j}\leq \frac{n}{\ln n}\), and \(t_{n}=\sqrt{\frac{\ln n}{n}}\) that
$$ \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n} \widetilde{\eta }_{i} \Biggr\vert \geq \frac{\kappa t_{n}}{4} \Biggr\} \lesssim \exp \biggl(- \frac{n(\kappa t_{n}/4)^{2}}{2(\sigma ^{2}+ \kappa t_{n}2^{j/2}/12)} \biggr)\lesssim \exp \biggl(-\frac{(\ln n) \kappa ^{2}/16}{2(\sigma ^{2}+\kappa /12)} \biggr). $$
Clearly, we can take \(\kappa >1\) such that \(\mathbb{P} \{ \vert \frac{1}{n} \sum_{i=1}^{n}\widetilde{\eta }_{i} \vert \geq \frac{\kappa t _{n}}{4} \} \lesssim 2^{-\lambda j}\). Then similar arguments show that \(\mathbb{P} \{ \vert \frac{1}{n}\sum_{i=1}^{n}\overline{ \eta }_{i} \vert \geq \frac{\kappa t_{n}}{4} \} \lesssim 2^{- \lambda j}\). Combining those with (18), we obtain
$$\begin{aligned} \mathbb{P} \Biggl\{ \Biggl\vert \frac{1}{n}\sum _{i=1}^{n} \biggl(\frac{ \mu \psi _{j,k}(Y_{i})}{\omega (Y_{i})}- \beta _{j,k} \biggr) \Biggr\vert \geq \frac{\kappa t_{n}}{2} \Biggr\} \lesssim 2^{-\lambda j}. \end{aligned}$$
(19)
By (16), (17), and (19) we get
$$ \mathbb{P} \bigl\{ \vert \widehat{\beta }_{j,k}-\beta _{j,k} \vert \geq \kappa t_{n} \bigr\} \lesssim 2^{-\lambda j}. $$
This ends the proof. □