A simultaneous confidence corridor for varying coefficient regression with sparse functional data

Abstract

We consider a varying coefficient regression model for sparse functional data, with time varying response variable depending linearly on some time-independent covariates with coefficients as functions of time-dependent covariates. Based on spline smoothing, we propose data-driven simultaneous confidence corridors for the coefficient functions with asymptotically correct confidence level. Such confidence corridors are useful benchmarks for statistical inference on the global shapes of coefficient functions under any hypotheses. Simulation experiments corroborate with the theoretical results. An example in CD4/HIV study is used to illustrate how inference is made with computable p values on the effects of smoking, pre-infection CD4 cell percentage and age on the CD4 cell percentage of HIV infected patients under treatment.

Appendix A

Throughout this section, $a_{n}\sim b_{n}$ means $\lim \nolimits _{n \rightarrow \infty }b_{n}/a_{n}=c$, where $c$ is some nonzero constant. For functions $a_{n}(t)$, $b_{n}(t)$, $a_{n}(t)={\scriptstyle {\mathcal {U}}} \left\{ b_{n}(t)\right\}$ means $a_{n}(t)/b_{n}(t)\rightarrow 0$ as $n\rightarrow \infty$ uniformly for $t\in \left[ 0,1\right]$, and $a_{n}(t)= \mathcal {U}\left\{ b_{n}(t)\right\}$ means $a_{n}(t)/b_{n}(t)=\mathcal {O} (1)$ as $n\rightarrow \infty$ uniformly for $t\in \left[ 0,1\right]$. We use ${\scriptstyle {\mathcal {U}}}_{p}(\cdot )$ and $\mathcal {U}_{p}(\cdot )$ if the convergence is in the sense of uniform convergence in probability.

1.1 A.1 Technical assumptions

We define the modulus of continuity of a continuous function $\phi$ on $\left[ a,b\right]$ by $\omega \left( \phi ,\delta \right) =\max \nolimits _{t,t^{\prime }\in \left[ a,b\right] ,\left| t-t^{\prime }\right| \le \delta }\left| \phi (t)-\phi \left( t^{\prime }\right) \right|$. For any $r\in \left( 0,1\right]$, denote the collection of order $r$ Hõlder continuous function on $\left[ 0,1\right]$ by

$$\begin{aligned} C^{0,r}\left[ 0,1\right] =\left\{ \phi :\Vert \phi \Vert _{0,r}=\sup _{t\ne t^{\prime },t,t^{\prime }\in \left[ 0,1\right] }\frac{\left| \phi (t)-\phi \left( t^{\prime }\right) \right| }{\left| t-t^{\prime }\right| ^{r}}<+\infty \right\} , \end{aligned}$$

in which $\left\| \phi \right\| _{0,r}$ is the $C^{0,r}$-seminorm of $\phi$. Let $C\left[ 0,1\right]$ be the collection of continuous function on $\left[ 0,1\right]$. Clearly, $C^{0,r}\left[ 0,1\right] \subset C\left[ 0,1\right]$ and, if $\phi \in C^{0,r}\left[ 0,1\right]$, then $\omega \left( \phi ,\delta \right) \le \left\| \phi \right\| _{0,r}\delta ^{r}$.

The following regularity assumptions are needed for the main results.

• (A1) The regression functions $m_{l}(t)\in C^{0,1}\left[ 0,1 \right]$, $l=1,\ldots ,d$.

• (A2) The set of random variables $\left( T_{ij},\varepsilon _{ij},N_{i},\xi _{ik,l},X_{il}\right) _{i=1,j=1,k=1,l=1}^{n,N_{i},\infty ,d}$ is a subset of variables $\left( T_{ij},\varepsilon _{ij},N_{i},\xi _{ik,l},X_{il}\right) _{i=1,j=1,k=1,l=1}^{\infty ,\infty ,\infty ,d}$ consisting of independent random variables, in which all $T_{ij}$’s i.i.d with $T_{ij}\sim T$, where $T$ is a random variable with probability density function $f(t)$; $X_{il}$’s i.i.d for each $l=1,\ldots ,d$; $N_{i}$’s i.i.d with $N_{i}\sim N$, where $N>0\$ is a positive integer-valued random variable with ${{\mathsf {E}}} \{N^{2r}\}\le r!c_{N}^{r}$, $r=2,3,\ldots$, for some constant $c_{N}>0$. Variables$\left( \xi _{ik,l}\right) _{i=1,k=1,l=1}^{\infty ,\infty ,d}$ and $\left( \varepsilon _{ij}\right) _{i=1,j=1}^{\infty ,\infty }$ are i.i.d $N\left( 0,1\right)$.

• (A3) The functions $f(t)$, $\sigma (t)$ and $\phi _{k,l}(t)\in C^{0,r}\left[ 0,1\right]$ for some $r\in \left( 0,1 \right]$ with $f(t)\in \left[ c_{f},C_{f}\right]$, $\sigma (t)\in \left[ c_{\sigma },C_{\sigma }\right]$, $t\in \left[ 0,1\right]$, for constants $0<c_{f}\le C_{f}<+\infty$, $0<c_{\sigma }\le C_{\sigma }<+\infty .$

• (A4) For $l=1,\ldots ,d$, $\sum _{k=1}^{\infty }\left\| \phi _{k,l}\right\| _{\infty }<+\infty$, and $G_{l}(t,t)\in \left[ c_{G,l},C_{G,l}\right]$, $t\in \left[ 0,1\right]$, for constants $0<c_{G,l}\le C_{G,l}<+\infty$.

• (A5) There exist constants $0<c_{\mathbf {H}}\le C_{\mathbf {H }}<+\infty$ and $0<c_{\eta }\le C_{\eta }<+\infty$, such that $c_{\mathbf {H}}I_{d\times d}\le \mathbf {H}=\{H_{ll^{\prime }}\}_{l,l^{\prime }=1}^{d}={{\mathsf {E}}}\left( \mathbf {XX}^{ \scriptstyle {\mathsf {T}}}\right) \le C_{\mathbf {H}}I_{d\times d}$. For some $\eta >4$, $l=1,\ldots ,d$, $c_{\eta }\le {{\mathsf {E}}}\left| X_{l}\right| ^{8+\eta }\le C_{\eta }$ .

• (A6) As $n\rightarrow \infty$, the number of interior knots $N_{{\mathrm{s}}}={\scriptstyle {\mathcal {O}}}\left( n^{\vartheta }\right)$ for some $\vartheta \in \left( 1/3,1/2\right)$ while $N_{{\mathrm{s}}}^{-1}={\scriptstyle {\mathcal {O}}}\left\{ n^{-1/3}\left( \log (n)\right) ^{-1/3}\right\}$. The subinterval length $h_{{\mathrm{s}}}\sim N_{ {\mathrm{s}}}^{-1}$.

Assumptions (A1)–(A3) are common conditions used in the literature; see for example, Ma et al. (2012). Assumption (A1) controls the rate of convergence of the spline approximation $\hat{m}_{l}$, $l=1,\ldots ,d$. The requirement of $N_{i}$ in Assumption (A2) ensures that the observation times are randomly scattered, reflecting sparse and irregular designs. Assumption (A4) guarantees that the random variable $\sum _{k=1}^{\infty }\xi _{ik,l}\phi _{k,l}(t)$ absolutely uniformly converges. Assumption (A5) is analog to Assumption (A2) in Liu and Yang (2010), ensuring that the $X_{il}$s are not multicollinear. Assumption (A6) describes the requirement of the growth rate of the dimension of the spline spaces relative to the sample size.

1.2 A.2 Preliminaries

Lemma 1

(Bosq (1998), Theorem 1.2). Suppose that $\left\{ \xi _{i}\right\} _{i=1}^{n}$ are i.i.d with ${{\mathsf {E}}}(\xi _{1})=0,\sigma ^{2}={{\mathsf {E}}}\xi _{1}^{2}$, and there exists $c>0$ such that for $r=3,4,\ldots$, ${{\mathsf {E}}}\left| \xi _{1}\right| ^{r}\le c^{r-2}r!{{\mathsf {E}}}\xi _{1}^{2}<+\infty$. Then for each $n>1$, $t>0$, $P(\left| S_{n}\right| \ge \sqrt{n} \sigma t)\le 2\exp \left( -t^{2}\left( 4+2ct/\sqrt{n}\sigma \right) ^{-1}\right)$, in which $S_{n}=\sum _{i=1}^{n}\xi _{i}.$

Lemma 2

Under Assumptions (A2)–(A6), we have

$$\begin{aligned} A_{n,1}=\underset{0\le J\le N_{{\mathrm{s}}},1\le l,l^{\prime }\le d}{ \sup }\frac{\left| \left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle _{N_{{\mathrm{T}}}}-\left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle \right| }{\sqrt{\left\langle B_{J}X_{l},B_{J}X_{l}\right\rangle }\sqrt{\left\langle B_{J}X_{l^{\prime }},B_{J}X_{l^{\prime }}\right\rangle }}=\mathcal {O}_{p}\left( \sqrt{\frac{ \log \left( n\right) }{nh_{{\mathrm{s}}}}}\right) , \end{aligned}$$

where for any $J=0,\ldots ,N_{{\mathrm{s}}}$ and $l,l^{\prime }=1,\ldots ,d$,

$$\begin{aligned} \left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle _{N_{{\mathrm{T}}}}&= N_{{\mathrm{T}}}^{-1}\sum \nolimits _{i=1}^{n}\sum \nolimits _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})X_{il}X_{il^{\prime }}, \\ \left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle&= {{\mathsf {E}}}\left\{ B_{J}^{2}(T_{ij})X_{il}X_{il^{\prime }}\right\} =H_{ll^{\prime }}. \end{aligned}$$

Proof

Let $\omega _{i,J}=\omega _{i,J,l,l^{\prime }}=\sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})X_{il}X_{il^{\prime }}$, then ${{\mathsf {E}}}\omega _{i,J}={{\mathsf {E}}}N_{1}H_{ll^{\prime }}\sim 1$ and ${{\mathsf {E}}}\left( \omega _{ij,J}\right) ^{2}= {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\right\} ^{2} {{\mathsf {E}}}\left( X_{il}X_{il^{\prime }}\right) ^{2} \sim h_{ {\mathrm{s}}}^{-1}$. Next define a sequence $D_{n}=n^{\alpha }$ with $\alpha (4+\eta /2)>1$ and $\sqrt{\log \left( n\right) }D_{n}n^{-1/2}h_{{\mathrm{s}} }^{-1/2}\rightarrow 0$, $n^{1/2}h_{{\mathrm{s}}}^{1/2}D_{n}^{-\left( 3+\eta /2\right) } \rightarrow 0$, which necessitates $\eta >2$ according to Assumption (A5). We make use of the following truncated and tail decomposition

$$\begin{aligned} X_{ill^{\prime }}=X_{il}X_{il^{\prime }}=X_{ill^{\prime },1}^{D_{n}}+X_{ill^{\prime },2}^{D_{n}}, \end{aligned}$$

where $X_{ill^{\prime },1}^{D_{n}}=X_{il}X_{il^{\prime }}I\left\{ \left| X_{il}X_{il^{\prime }}\right| >D_{n}\right\}$, $X_{ill^{\prime },2}^{D_{n}}=X_{il}X_{il^{\prime }}I\left\{ \left| X_{il}X_{il^{\prime }}\right| \le D_{n}\right\}$. Correspondingly, the truncated and tail parts of $\omega _{i,J}$ are $\omega _{i,J,m}=B_{J}^{2}(T_{ij})X_{ill^{\prime },m}^{D_{n}}$, $m=1,2$. According to Assumption (A5), for any $l,l^{\prime }=1,\ldots ,d$,

$$\begin{aligned} \sum _{n=1}^{\infty }P\left\{ \left| X_{nl}X_{nl^{\prime }}\right| >D_{n}\right\} \le \sum _{n=1}^{\infty }\frac{{{\mathsf {E}}} \left| X_{nl}X_{nl^{\prime }}\right| ^{4+\eta /2}}{D_{n}^{4+\eta /2}} \le C_{\eta }\sum _{n=1}^{\infty }D_{n}^{-\left( 4+\eta /2\right) }<\infty . \end{aligned}$$

By Borel–Cantelli Lemma, one has $\sum \nolimits _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})X_{ill^{\prime },1}^{D_{n}}=0,a.s.$. So we obtain

$$\begin{aligned} \sup _{J,l,l^{\prime }}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,1}\right| =\mathcal {O}_{a.s.}\left( n^{-k}\right) , ~~k\ge 1, \end{aligned}$$

and

$$\begin{aligned} {{\mathsf {E}}}\omega _{i,J,1}&= {{\mathsf {E}}}\left( X_{ill^{\prime },1}^{D_{n}}\right) {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\right\} \\&\le D_{n}^{-\left( 3+\eta /2\right) }{{\mathsf {E}}}\left| X_{il}X_{il^{\prime }}\right| ^{4+\eta /2}{{\mathsf {E}}}N_{1} {{\mathsf {E}}}B_{J}^{2}(T_{ij})\le cD_{n}^{-\left( 3+\eta /2\right) }. \end{aligned}$$

Next we considerate the truncated part $\omega _{i,J,2}$. For large $n$, ${{\mathsf {E}}}\left( \omega _{i,J,2}\right) ={{\mathsf {E}}} \left( \omega _{i,J}\right) -{{\mathsf {E}}}\left( \omega _{i,J,1}\right) \sim 1$, ${{\mathsf {E}}}\left( \omega _{i,J,2}\right) ^{2}={{\mathsf {E}}}\left( \omega _{i,J}\right) ^{2}- {{\mathsf {E}}}\left( \omega _{i,J,1}\right) ^{2}\sim h_{{\mathrm{s}}}^{-1}$. Define $\omega _{i,J,2}^{*}=\omega _{i,J,2}-{{\mathsf {E}}}\left( \omega _{i,J,2}\right)$, then ${{\mathsf {E}}}\omega _{i,J,2}^{*}=0$, and

$$\begin{aligned} {{\mathsf {E}}}\left( \omega _{i,J,2}^{*}\right) ^{2}&= {{\mathsf {E}}}\left( \omega _{i,J,2}\right) ^{2}-\left( {{\mathsf {E}}}\omega _{i,J,2}\right) ^{2}={{\mathsf {E}}} \left\{ \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})X_{ill^{\prime },2}^{D_{n}}\right\} ^{2}-\mathcal {U}\left( 1\right) \\&= {{\mathsf {E}}}\left( X_{ill^{\prime },2}^{D_{n}}\right) ^{2} {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\right\} ^{2}-\mathcal {U}\left( 1\right) . \end{aligned}$$

Note that

$$\begin{aligned} {{\mathsf {E}}}\left( X_{ill^{\prime },2}^{D_{n}}\right) ^{2} {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\right\} ^{2}&\ge \left\{ {{\mathsf {E}}}\left( X_{ill^{\prime }}\right) ^{2}-{{\mathsf {E}}}\left( X_{ill^{\prime },1}^{D_{n}}\right) ^{2}\right\} {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{4}(T_{ij})\right\} \\&\ge \left\{ {{\mathsf {E}}}\left( X_{ill^{\prime }}\right) ^{2}-{ \scriptstyle {\mathcal {U}}}\left( 1\right) \right\} {{\mathsf {E}}}N_{1} {{\mathsf {E}}}B_{J}^{4}(T_{ij}). \end{aligned}$$

Thus, there exists $c_{\omega }$ such that for large $n$, ${{\mathsf {E}}}\left( \omega _{i,J,2}^{*}\right) ^{2}\ge c_{\omega } {{\mathsf {E}}}\left( X_{ill^{\prime }}\right) ^{2}h_{{\mathrm{s}}}^{-1}$. Next for any $r>2$

$$\begin{aligned}&{{\mathsf {E}}}\left| \omega _{i,J,2}^{*}\right| ^{r}= {{\mathsf {E}}}\left| \omega _{i,J,2}-{{\mathsf {E}}}\left( \omega _{i,J,2}\right) \right| ^{r}\le 2^{r-1}\left( {{\mathsf {E}}}\left| \omega _{i,J,2}\right| ^{r}+\left| {{\mathsf {E}}}\left( \omega _{i,J,2}\right) \right| ^{r}\right) \\&\quad = 2^{r-1}\left\{ {{\mathsf {E}}}\left| X_{ill^{\prime },2}^{D_{n}}\right| ^{r}{{\mathsf {E}}}\left| \sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\right| ^{r}+\mathcal {U}(1)\right\} \\&\quad =2^{r-1}\!\left[ \!{{\mathsf {E}}}\left| X_{ill^{\prime },2}^{D_{n}}\right| ^{r}{{\mathsf {E}}}\left\{ \,\sum _{0\le \, r_{1},\ldots ,r_{N_{i}}\le \, r}^{r_{1}+\cdots + r_{N_{i}} =\, r}\left( {\begin{array}{c}r\\ r_{1}\cdots r_{N_{i}}\end{array}}\right) \prod \limits _{j=1}^{N_{i}}{{\mathsf {E}}} B_{J}^{2r_{j}}\left( T_{ij}\right) \right\} \!\!+\!\mathcal {U}(1)\!\right] \!, \end{aligned}$$

then there exists $C_{\omega }>0$ such that for any $r>2$ and large $n$,

$$\begin{aligned} {{\mathsf {E}}}\left| \omega _{i,J,2}^{*}\right| ^{r}&\le 2^{r-1}\left[ D_{n}^{r-2}{{\mathsf {E}}}\left( X_{ill^{\prime }}\right) ^{2}{{\mathsf {E}}}\left\{ N_{1}^{r}\max \prod \limits _{j=1}^{N_{i}}{{\mathsf {E}}}B_{J}^{2r_{j}}\left( T_{ij}\right) \right\} +\mathcal {U}(1)\right] \\&\le 2^{r-1}\left[ D_{n}^{r-2}{{\mathsf {E}}}\left( X_{ill^{\prime }}\right) ^{2}\left( {{\mathsf {E}}}N_{1}^{r}\right) C_{B}h_{{\mathrm{s }}}^{1-r}+\mathcal {U}(1)\right] \\&\le 2^{r}D_{n}^{r-2}\left( c_{N}^{r}r!\right) ^{1/2}C_{B}h_{{\mathrm{s}} }^{2-r}c_{\omega }^{-1}{{\mathsf {E}}}\left( \omega _{i,J,2}^{*}\right) ^{2} \\&\le \left( C_{\omega }D_{n}h_{{\mathrm{s}}}^{-1}\right) ^{r-2}r! {{\mathsf {E}}}\left( \omega _{i,J,2}^{*}\right) ^{2}, \end{aligned}$$

which implies that $\left\{ \omega _{i,J,2}^{*}\right\} _{i=1}^{n}$ satisfies Cramér’s condition with constant $C_{\omega }D_{n}h_{{\mathrm{s}} }^{-1}$. Applying Lemma 1 to $\sum _{i=1}^{n}\omega _{i,J,2}^{*}$, for $r>2$ and any large enough $\delta >0$, $P\left\{ \left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,2}^{*}\right| \ge \delta \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\}$ is bounded by

$$\begin{aligned} 2\exp \left\{ \frac{-\delta ^{2}\left( \log (n)\right) }{4+2C_{\omega }D_{n}h_{{\mathrm{s}}}^{-1}\delta \left( \log (n)\right) ^{1/2}n^{-1/2}h_{{\mathrm{s }}}^{1/2}}\right\} \le 2n^{-8}. \end{aligned}$$

Hence

$$\begin{aligned} \sum _{n=1}^{\infty }P\left\{ \sup _{0\le J\le N_{{\mathrm{s}}},1\le l,l^{\prime }\le d}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,2}^{*}\right| \ge \delta \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} <\infty . \end{aligned}$$

Thus, $\sup _{J,l,l^{\prime }}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,2}^{*}\right| =\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}} }\right) ^{-1/2}(\log (n))^{1/2}\right\}$ as $n\rightarrow \infty$ by Borel–Cantelli Lemma. Furthermore,

$$\begin{aligned}&\sup _{J,l,l^{\prime }}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J}- {{\mathsf {E}}}\omega _{i,J}\right| \\&\qquad \le \sup _{J,l,l^{\prime }}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,1}\right| +\sup _{J,l,l^{\prime }}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J,2}^{*}\right| +\sup _{J,l,l^{\prime }}\left| {{\mathsf {E}}}\omega _{i,J,1}\right| \\&\qquad =\mathcal {U}_{a.s.}\left( n^{-k}\right) +\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} +\mathcal {U} \left( D_{n}^{-\left( 3+\eta /2\right) }\right) \\&\qquad =\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

Finally, we notice that

$$\begin{aligned}&\sup _{J,l,l^{\prime }}\left| \left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle _{N_{{\mathrm{T}}}}\!-\!\left\langle B_{J}X_{l},B_{J}X_{l^{\prime }}\right\rangle \right| \!=\!\sup _{J,l,l^{\prime }}\left| \left( nN_{ {\mathrm{T}}}^{-1}\right) n^{-1}\sum _{i=1}^{n}\omega _{i,J}-\left( {{\mathsf {E}}}N_{1}\right) ^{-1}{{\mathsf {E}}}\omega _{i,J}\right| \\&\quad \le \! \sup _{J,l,l^{\prime }}\left( {{\mathsf {E}}}N_{1}\right) ^{-1}\left| \left( n{{\mathsf {E}}}N_{1}\right) N_{{\mathrm{T}} }^{-1}\!-\!1\right| \left| n^{-1}\sum _{i=1}^{n}\omega _{i,J}\right| \!+\!\sup _{J,l,l^{\prime }}\left( {{\mathsf {E}}}N_{1}\right) ^{-1}\left| n^{-1}\sum _{i=1}^{n}\omega _{i,J}\!-\!{{\mathsf {E}}} \omega _{i,J}\right| \\&\quad =\mathcal {O}_{p}\left( n^{-1/2}\right) +\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} =\mathcal {O} _{p}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} , \end{aligned}$$

and $\left\langle B_{J}X_{l},B_{J}X_{l}\right\rangle =H_{ll}=\mathcal {U}(1)$. Hence, $A_{n,1}=\mathcal {O}_{p}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\}$. $\square$

For the random matrix $\hat{\mathbf {V}}_{J}$ defined in (18), the lemma below shows that its inverse can be approximated by the inverse of a deterministic matrix $\mathbf {H}={{\mathsf {E}}}(\mathbf {XX}^{ \scriptstyle {\mathsf {T}}})$.

Lemma 3

Under Assumptions (A2) and (A4)–(A6), for any $J=0,\ldots , N_{{\mathrm{s}}}$, we have

$$\begin{aligned} \hat{\mathbf {V}}_{J}^{-1}=\mathbf {H}^{-1}+\mathcal {O}_{p}\left\{ \left( nh_{{\mathrm{s}} }\right) ^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

(25)

Proof

By Lemma 2, we have

$$\begin{aligned} \left\| \hat{\mathbf {V}}_{J}-\mathbf {H}\right\| _{\infty }=\mathcal {O} _{p}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

Using the fact that for any matrices $\mathbf {A}$ and $\mathbf {B}$,

$$\begin{aligned} \left( \mathbf {A}+h\mathbf {B}\right) ^{-1}=\mathbf {A}^{-1}-h\mathbf {A}^{-1} \mathbf {B}\mathbf {A}^{-1}+\mathcal {O}(h^{2}), \end{aligned}$$

we obtain (25). $\square$

The next lemma implies that the difference between $\tilde{\varvec{\xi }} \left( t\right)$ and $\hat{\varvec{\xi }}\left( t\right)$ and the difference between $\tilde{\varvec{\varepsilon }}(t)$ and $\hat{ \varvec{\varepsilon }}(t)$ are both negligible uniformly over $t\in \left[ 0,1\right]$.

Lemma 4

Under Assumption (A2)–(A6), for $\tilde{\varvec{ \xi }}(t)$, $\tilde{\varvec{\varepsilon }}(t)$ given in (36), (37) and $\hat{\varvec{\xi }}(t)$, $\hat{\varvec{\varepsilon } }(t)$ given in (38), (39), as $n\rightarrow \infty$, we have

$$\begin{aligned} \sup _{t\in \left[ 0,1\right] }\left\| \tilde{\varvec{\xi }}\left( t\right) -\hat{\varvec{\xi }}\left( t\right) \right\| _{\infty }&= \mathcal {O}_{p}\left\{ n^{-1}h_{{\mathrm{s}}}^{-3/2}\log (n)\right\} , \end{aligned}$$

(26)

$$\begin{aligned} \sup _{t\in \left[ 0,1\right] }\left\| \tilde{\varvec{\varepsilon }} \left( t\right) -\hat{\varvec{\varepsilon }}\left( t\right) \right\| _{\infty }&= \mathcal {O}_{p}\left\{ n^{-1}h_{{\mathrm{s}}}^{-3/2} \log (n)\right\} . \end{aligned}$$

(27)

Proof

Comparing the equations of $\tilde{\varvec{\xi }}(t)$ and $\hat{\varvec{\xi }}(t)$ given in (A.2) and (A.4), we let

$$\begin{aligned} \frac{1}{N_{{\mathrm{T}}}}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J}(T_{ij})X_{il} \sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\xi _{ik,l^{\prime \prime }}\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) X_{il^{\prime \prime }}= \frac{n}{N_{{\mathrm{T}}}}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\Omega _{i,J,l^{\prime \prime },l}. \end{aligned}$$

where $\Omega _{i,J,l^{\prime \prime },l}=\Omega _{i}=n^{-1}\left[ X_{il}X_{il^{\prime \prime }}\sum _{k=1}^{\infty }\left\{ \sum _{j=1}^{N_{i}}B_{J}(T_{ij})\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} \xi _{ik,l^{\prime \prime }}\right]$. Note that ${{\mathsf {E}}}\Omega _{i}=0$ and

$$\begin{aligned} \sigma _{\Omega _{i},n}^{2}&= {{\mathsf {E}}}\left( \Omega _{i}^{2}\left| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right. \right) \\&= n^{-2}\left[ X_{il}X_{il^{\prime \prime }}\sum _{k=1}^{\infty }\left\{ \sum _{j=1}^{N_{i}}B_{J}(T_{ij})\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} ^{2}\right] \\&\le n^{-2}\left\{ X_{il}^{2}X_{il^{\prime \prime }}^{2}\sum _{k=1}^{\infty }N_{i}\sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})\phi _{k,l^{\prime \prime }}^{2}\left( T_{ij}\right) \right\} \\&= n^{-2}\left\{ X_{il}^{2}X_{il^{\prime \prime }}^{2}N_{i}\sum _{j=1}^{N_{i}}B_{J}^{2}(T_{ij})G_{l^{\prime \prime }}\left( T_{ij},T_{ij}\right) \right\} \\&\le Cn^{-2}h_{{\mathrm{s}}}^{-1}X_{il}^{2}X_{il^{\prime \prime }}^{2}N_{i}^{2}. \end{aligned}$$

Given $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$, $\left\{ \sigma _{\Omega _{i},n}^{-1}\Omega _{i}\right\} _{i=1}^{n}$ are i.i.d $N\left( 0,1\right)$. It is easy to show that for any large enough $\delta >0$,

$$\begin{aligned}&P\left\{ \frac{\left| \sum _{i=1}^{n}\Omega _{i}\right| }{\sqrt{ \sum _{i=1}^{n}\sigma _{\Omega _{i},n}^{2}}}\ge \left. \delta \sqrt{\log (n)} \right| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right\} \\&\quad \le 2\exp \left\{ -\frac{1}{2}\delta ^{2}\log (n)\right\} \le 2n^{-8}, \end{aligned}$$

$$\begin{aligned} P\left[ \!\left| \!\sum _{i=1}^{n}\!\Omega _{i}\right| \!\ge \! \left. \!\delta \! \left\{ \frac{C\log (n)}{nh_{{\mathrm{s}}}}n^{-1}\!\sum _{i=1}^{n}X_{il}^{2}X_{il^{ \prime \prime }}^{2}N_{i}^{2}\right\} ^{1/2}\right| \!\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right] \!\le \! 2n^{-8}. \end{aligned}$$

Note that $n^{-1}\sum _{i=1}^{n}X_{il}^{2}X_{il^{\prime \prime }}^{2}N_{i}^{2}=\mathcal {O}_{p}\left( 1\right)$, hence

$$\begin{aligned} \sum _{n=1}^{\infty }P\left\{ \sup _{0\le J\le N_{{\mathrm{s}}},1\le l,l^{\prime \prime }\le d}\left| \sum _{i=1}^{n}\Omega _{i,J,l^{\prime \prime },l}\right| \ge \delta \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} <\infty . \end{aligned}$$

Thus, $\sup _{J,l,l^{\prime \prime }}\left| \sum _{i=1}^{n}\Omega _{i,J,l^{\prime \prime },l}\right| =\mathcal {O}_{p}\left\{ \left( nh_{ {\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\}$ as $n\rightarrow \infty$ by Borel–Cantelli Lemma. It follows that $\sup _{J,l}\left| nN_{ {\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\Omega _{i,J,l^{\prime \prime },l}\right| = \mathcal {O}_{p}\big \{\left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\big \}$. Finally, according to Lemma 25, we obtain (26). (27) is proved similarly. $\square$

Denote the inverse matrix of $\mathbf {H}$ by $\mathbf {H}^{-1}=\{z_{ll^{ \prime }}\}_{l,l^{\prime }=1}^{d}$. For any $l=1,\ldots ,d$, we rewrite the $l$th element of $\hat{\xi }_{l}(t)$ and $\hat{\varepsilon }_{l}(t)$ in (38) and (39) as the following

$$\begin{aligned} \hat{\xi }_{l}(t)&= c_{J(t),n}^{-1/2}N_{{\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J(t),l^{\prime \prime },l}\xi _{ik,l^{\prime \prime }}, \end{aligned}$$

(28)

$$\begin{aligned} \hat{\varepsilon }_{l}(t)&= c_{J(t),n}^{-1/2}N_{{\mathrm{T}} }^{-1}\sum _{i=1}^{n}\sum _{j=1}^{N}R_{ij,\varepsilon ,J(t),l}\varepsilon _{ij}, \end{aligned}$$

(29)

where for any $0\le J\le N_{{\mathrm{s}}}$,

$$\begin{aligned}&\displaystyle R_{ik,\xi ,J,l^{\prime \prime },l}=\left( \sum _{l^{\prime }=1}^{d}z_{ll^{\prime }}X_{il^{\prime }}X_{il^{\prime \prime }}\right) \left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} , \end{aligned}$$

(30)

$$\begin{aligned}&\displaystyle R_{ij,\varepsilon ,J,l}=\left( \sum _{l^{\prime }=1}^{d}z_{ll^{\prime }}X_{il^{\prime }}\right) B_{J}\left( T_{ij}\right) \sigma \left( T_{ij}\right) . \end{aligned}$$

(31)

Further denote

$$\begin{aligned} S_{ill^{\prime \prime }}=\left( \sum _{l^{\prime }=1}^{d}z_{ll^{\prime }}X_{il^{\prime }}X_{il^{\prime \prime }}\right) ^{2},\quad s_{ll^{\prime \prime }}={{\mathsf {E}}}\left( S_{ill^{\prime \prime }}\right) ,\quad 1\le l,l^{\prime \prime }\le d. \end{aligned}$$

(32)

Lemma 5

Under Assumptions (A2)–(A6), for $R_{ik,\xi ,J,l^{\prime \prime },l},$ $R_{ij,\varepsilon ,J,l}$ in (30), (31),

$$\begin{aligned}&{{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}\right) =c_{J,n}^{-1}s_{ll^{\prime \prime }}\left[ \left( {{\mathsf {E}}}N_{1}\right) \int \nolimits b_{J}\left( u\right) G_{l^{\prime \prime }}\left( u,u\right) f\left( u\right) \mathrm{d}u\right. \\&\quad \left. +{{\mathsf {E}}}\left\{ N_{1}(N_{1}-1)\right\} \int b_{J}\left( u\right) b_{J}\left( v\right) G_{l^{\prime \prime }}\left( u,v\right) f\left( u\right) f\left( v\right) \mathrm{d}u\mathrm{d}v\!\right] \!, \end{aligned}$$

$$\begin{aligned} {{\mathsf {E}}}R_{ij,\varepsilon ,J,l}^{2}=c_{J,n}^{-1}z_{ll}\int b_{J}\left( u\right) \sigma ^{2}\left( u\right) f\left( u\right) \mathrm{d}u, \end{aligned}$$

for $0\le J\le N_{{\mathrm{s}}}$ and $0\le l,l^{\prime \prime }\le d$. In addition, there exist $0<c_{R}<C_{R}<\infty$, such that

$$\begin{aligned} c_{R}s_{ll^{\prime \prime }} \le {{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}\right) \le C_{R}s_{ll^{\prime \prime }},\quad c_{R}\le {{\mathsf {E}}}R_{ij,\varepsilon ,J,l}^{2} \le C_{R}, \end{aligned}$$

for $0\le J\le N_{{\mathrm{s}}}$, $0\le l,l^{\prime \prime }\le d$, and as$\ n\rightarrow \infty$

$$\begin{aligned} A_{n,\xi }&= \sup _{J,l^{\prime \prime },l}\left| n^{-1}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}-{{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}\right) \right| \\&= \mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} , \\ A_{n,\varepsilon }&= \sup _{J,l}\left| N_{{\mathrm{T}}}^{-1}\sum _{i=1}^{n} \sum _{j=1}^{N_{i}}R_{ij,\varepsilon ,J,l}^{2}-{{\mathsf {E}}}R_{ij,\varepsilon ,J,l}^{2}\right| =\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}( \log (n))^{1/2}\right\} . \end{aligned}$$

Proof

By independence of $\left\{ T_{ij}\right\} _{j=1}^{\infty }$, $\left\{ X_{il}\right\} _{l=1}^{d},N_{i}$, the definition of $B_{J}$ and (32),

$$\begin{aligned}&{{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}\right) ={{\mathsf {E}}}\left( S_{ill^{\prime \prime }}\right) {{\mathsf {E}}}\sum _{k=1}^{\infty }\left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} ^{2} \\&\quad =s_{ll^{\prime \prime }}{{\mathsf {E}}} \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \\&\quad =s_{ll^{\prime \prime }}c_{J,n}^{-1}\left\{ \left( {{\mathsf {E}}}N_{1}\right) \int b_{J}\left( u\right) G_{l^{\prime \prime }}\left( u,u\right) f\left( u\right) \mathrm{d}u\right. \\&\left. \quad \quad +{{\mathsf {E}}}\left\{ N_{1}(N_{1}-1)\right\} \int b_{J}\left( u\right) b_{J}\left( v\right) G_{l^{\prime \prime }}\left( u,v\right) f\left( u\right) f\left( v\right) \mathrm{d}u\mathrm{d}v\right\} , \end{aligned}$$

thus there exist constants $0<c_{R}<C_{R}<\infty$ such that $c_{R}s_{ll^{\prime \prime }}\le {{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{1k,\xi ,J,l^{\prime \prime },l}^{2}\right) \le C_{R}s_{ll^{\prime \prime }}$, $0\le J\le N_{{\mathrm{s}}}$, $0\le l,l^{\prime \prime }\le d$.

If $s_{ll^{\prime \prime }}=0$, one has $S_{ill^{\prime \prime }}=0$, almost surely. Hence $n^{-1}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}=0$, almost surely. In the case of $s_{ll^{\prime \prime }}>0$, let $\zeta _{i,J}=\zeta _{i,J,l^{\prime \prime },l}=\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}^{2}$ for brevity. Under Assumption (A5), it is easy to verify that

$$\begin{aligned} 0\!<\!s_{ll^{\prime \prime }}^{2}\!\le \! {{\mathsf {E}}}\left( S_{ill^{\prime \prime }}\right) ^{2} \!&\le \!d^{3}\sum _{l^{\prime }\!=\!1}^{d}\! {{\mathsf {E}}}\left| z_{ll^{\prime }}X_{il^{\prime }}X_{il^{\prime \prime }}\right| ^{4} \!\le \! d^{3}\sum _{l^{\prime }\!=\!1}^{d}z_{ll^{\prime }}\left\{ {{\mathsf {E}}}\left| X_{il^{\prime }}\right| ^{8} {{\mathsf {E}}}\left| X_{il^{\prime \prime }}\right| ^{8}\right\} ^{1/2}\!<\!\infty . \end{aligned}$$

So for large $n$,

$$\begin{aligned} {{\mathsf {E}}}\left( \zeta _{i,J}\right) ^{2}&= {{\mathsf {E}}} \left\{ \left( S_{ill^{\prime \prime }}\right) ^{2}\left( \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \right) ^{2}\right\} \\&\ge {{\mathsf {E}}}\left( S_{ill^{\prime \prime }}\right) ^{2}\frac{ 1}{4}c_{G,l^{\prime \prime }}^{2}{{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \right\} ^{4}\ge c {{\mathsf {E}}}\sum _{j=1}^{N_{i}}B_{J}^{4}\left( T_{ij}\right) \ge ch_{{\mathrm{s}}}^{-1}, \end{aligned}$$

and

$$\begin{aligned} {{\mathsf {E}}}\left( \zeta _{i,J}\right) ^{2}&\le {{\mathsf {E}}}\left( S_{ill^{\prime \prime }}\right) ^{2}4C_{G,l^{\prime \prime }}^{2}{{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \right\} ^{4} \\&\le c{{\mathsf {E}}}\left[ \left. N_{1}^{3}\sum _{j=1}^{N_{i}} {{\mathsf {E}}}B_{J}^{4}\left( T_{ij}\right) \right| N_{1}\right] \le c{{\mathsf {E}}}N_{1}^{4}{{\mathsf {E}}}B_{J}^{4}\left( T_{ij}\right) \le ch_{{\mathrm{s}}}^{-1}. \end{aligned}$$

Define a sequence $D_{n}=n^{\alpha }$ that satisfies $\alpha \left( 2+\eta /4\right) >1$, $D_{n}n^{-1/2}h_{{\mathrm{s}}}^{-1/2}(\log (n))^{1/2} \rightarrow 0$, $n^{1/2}h_{{\mathrm{s}}}^{1/2}D_{n}^{-\left( 1+\eta /4\right) }\rightarrow 0$, which requires $\eta >4$ provided by Assumption (A5). We make use of the following truncated and tail decomposition

$$\begin{aligned} S_{ill^{\prime \prime }}=\sum _{l^{\prime }=1}^{d}\sum _{l^{\prime \prime \prime }=1}^{d}z_{ll^{\prime }}z_{ll^{\prime \prime \prime }}X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}=S_{ill^{\prime \prime },1}^{D_{n}}+S_{ill^{\prime \prime },2}^{D_{n}}, \end{aligned}$$

where

$$\begin{aligned} S_{ill^{\prime \prime },1}^{D_{n}}&= \sum _{l^{\prime }=1}^{d}\sum _{l^{\prime \prime \prime }=1}^{d}z_{ll^{\prime }}z_{ll^{\prime \prime \prime }}X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}I\left\{ \left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\right| >D_{n}\right\} , \\ S_{ill^{\prime \prime },2}^{D_{n}}&= \sum _{l^{\prime }=1}^{d}\sum _{l^{\prime \prime \prime }=1}^{d}z_{ll^{\prime }}z_{ll^{\prime \prime \prime }}X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}I\left\{ \left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\right| \le D_{n}\right\} . \end{aligned}$$

Define correspondingly the truncated and tail parts of $\zeta _{i,J}$ as

$$\begin{aligned} \zeta _{i,J,m}=S_{ill^{\prime \prime },m}^{D_{n}}\sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) ,\quad m=1,2. \end{aligned}$$

According to Assumption (A5), for any $l^{\prime },l^{\prime \prime },l^{\prime \prime \prime }=1,\ldots ,d$,

$$\begin{aligned} \sum _{n=1}^{\infty }P\left\{ \left| X_{nl^{\prime }}\!X_{nl^{\prime \prime \prime }}X_{nl^{\prime \prime }}^{2}\right| >D_{n}\right\} \!\le \! \!\sum _{n=1}^{\infty }\frac{{{\mathsf {E}}}\left| X_{nl^{\prime }}X_{nl^{\prime \prime \prime }}X_{nl^{\prime \prime }}^{2}\right| ^{2+\eta /4}}{D_{n}^{2+\eta /4}}\!\le \! C_{\eta }\!\sum _{n=1}^{\infty }D_{n}^{-\left( 2\!+\!\eta /4\right) }\!<\!\infty . \end{aligned}$$

Borel–Cantelli Lemma implies that

$$\begin{aligned}&\displaystyle P\left\{ \omega \left| \exists N\left( \omega \right) ,\left| X_{nl^{\prime }}X_{nl^{\prime \prime \prime }}X_{nl^{\prime \prime }}^{2}\left( \omega \right) \right| \le D_{n}\text { for }n>N\left( \omega \right) \right. \right\} =1,\\&\displaystyle P\left\{ \omega \left| \exists N\left( \omega \right) ,\left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\left( \omega \right) \right| \le D_{n},i=1,\ldots ,n\text { for } n>N\left( \omega \right) \right. \right\} =1,\\&\displaystyle P\left\{ \omega \left| \exists N\left( \omega \right) ,I\left\{ \left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\left( \omega \right) \right| \!>\!D_{n}\right\} \!=\!0,i\!=\!1,\ldots ,n\text { for }n>N\left( \omega \right) \right. \right\} =1. \end{aligned}$$

Furthermore, one has

$$\begin{aligned} n^{-1}\sum _{i=1}^{n}\left\{ S_{ill^{\prime \prime },1}^{D_{n}}\sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \right\} =0,\quad a.s. \end{aligned}$$

Therefore, one has

$$\begin{aligned} \sup _{J,l,l^{\prime \prime }}\left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,1}\right| =\mathcal {O}_{a.s.}\left( n^{-k}\right) ,\quad k\ge 1. \end{aligned}$$

Notice that

$$\begin{aligned} {{\mathsf {E}}}\left( S_{ill^{\prime \prime },1}^{D_{n}}\right)&= {{\mathsf {E}}}\left[ \sum _{l^{\prime }=1}^{d}\sum _{l^{\prime \prime \prime }=1}^{d}z_{ll^{\prime }}z_{ll^{\prime \prime \prime }}X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}I\left\{ \left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\right| >D_{n}\right\} \right] \\&\le D_{n}^{-\left( 1+\eta /4\right) }\sum _{l^{\prime }=1}^{d}\sum _{l^{\prime \prime \prime }=1}^{d}z_{ll^{\prime }}z_{ll^{\prime \prime \prime }}{{\mathsf {E}}}\left| X_{il^{\prime }}X_{il^{\prime \prime \prime }}X_{il^{\prime \prime }}^{2}\right| ^{2+\eta /4} \\&\le cD_{n}^{-\left( 1+\eta /4\right) }. \end{aligned}$$

So for large $n$,

$$\begin{aligned} {{\mathsf {E}}}\left( \zeta _{i,J,1}\right)&= {{\mathsf {E}}} \left( S_{ill^{\prime \prime },1}^{D_{n}}\right) {{\mathsf {E}}} \left\{ \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \right\} \\&\le cD_{n}^{-\left( 1+\eta /4\right) }2C_{G,l^{\prime \prime }} {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \right\} ^{2} \\&\le cD_{n}^{-\left( 1+\eta /4\right) }{{\mathsf {E}}}\left( N_{1}^{2}\right) {{\mathsf {E}}}B_{J}^{2}\left( T_{ij}\right) \\&\le cD_{n}^{-\left( 1+\eta /4\right) }. \end{aligned}$$

Next we considerate the truncated part $\zeta _{i,J,2}$. For large $n$, ${{\mathsf {E}}}\left( \zeta _{i,J,2}\right) ={{\mathsf {E}}} \left( \zeta _{i,J}\right) -{{\mathsf {E}}}\left( \zeta _{i,J,1}\right) \sim 1$, ${{\mathsf {E}}}\left( \zeta _{i,J,2}\right) ^{2}={{\mathsf {E}}}\left( \zeta _{i,J}\right) ^{2}- {{\mathsf {E}}}\left( \zeta _{i,J,1}\right) ^{2}\sim h_{{\mathrm{s}}}^{-1}$. Define $\zeta _{i,J,2}^{*}=\zeta _{i,J,2}-{{\mathsf {E}}}\left( \zeta _{i,J,2}\right)$, then ${{\mathsf {E}}}\zeta _{i,J,2}^{*}=0$, and there exist $c_{\zeta },C_{\zeta }>0$ such that for $r>2$ and large $n$,

$$\begin{aligned} {{\mathsf {E}}}\left( \zeta _{i,J,2}^{*}\right) ^{2}&= {{\mathsf {E}}}\left| S_{ill^{\prime \prime },2}^{D_{n}}\right| ^{2}{{\mathsf {E}}}\left| \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \right| ^{2}-\left( {{\mathsf {E}}} \zeta _{i,J,2}\right) ^{2} \\&\ge \left\{ {{\mathsf {E}}}\left| S_{ill^{\prime \prime }}\right| ^{2}-{{\mathsf {E}}}\left| S_{ill^{\prime \prime },1}^{D_{n}}\right| ^{2}\right\} \frac{1}{4}c_{G,l^{\prime \prime }}^{2} {{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}\left( T_{ij}\right) \right\} ^{4}-\mathcal {U}(1) \\&\ge \left\{ {{\mathsf {E}}}\left| S_{ill^{\prime \prime }}\right| ^{2}-{\scriptstyle {\mathcal {U}}}(1)\right\} \frac{1}{4} c_{G,l^{\prime \prime }}^{2}{{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}^{4}\left( T_{ij}\right) \right\} -\mathcal {U}(1) \\&\ge \frac{1}{2}{{\mathsf {E}}}\left| S_{ill^{\prime \prime }}\right| ^{2}\frac{1}{4}c_{G,l^{\prime \prime }}^{2} {{\mathsf {E}}}N_{1}{{\mathsf {E}}}B_{J}^{4}\left( T_{ij}\right) -\mathcal {U}(1) \\&\ge c_{\zeta }{{\mathsf {E}}}\left| S_{ill^{\prime \prime }}\right| ^{2}h_{{\mathrm{s}}}^{-1}, \end{aligned}$$

and

$$\begin{aligned} {{\mathsf {E}}}\left| \zeta _{i,J,2}^{*}\right| ^{r}&= {{\mathsf {E}}}\left| \zeta _{i,J,2}-{{\mathsf {E}}}\left( \zeta _{i,J,2}\right) \right| ^{r}\le 2^{r-1}\left( {{\mathsf {E}}}\left| \zeta _{i,J,2}\right| ^{r}\!+\!\left| {{\mathsf {E}}} \left( \zeta _{i,J,2}\right) \right| ^{r}\right) \\&= 2^{r-1}\!\left\{ {{\mathsf {E}}}\left| S_{ill^{\prime \prime },2}^{D_{n}}\right| ^{r}{{\mathsf {E}}}\!\left| \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \right| ^{r}\!+\!\mathcal {U}(1)\right\} \\&\le 2^{r-1}\left[ \left( cD_{n}\right) ^{r-2}{{\mathsf {E}}} \left| S_{ill^{\prime \prime }}\right| ^{2}\left( 2C_{G,l^{\prime \prime }}\right) ^{r}{{\mathsf {E}}}\left\{ \sum _{j=1}^{N_{i}}B_{J}(T_{ij})\right\} ^{2r}+\mathcal {U}(1)\right] \\&\le 2^{r-1}\left[ \left( cD_{n}\right) ^{r-2}{{\mathsf {E}}} \left| S_{ill^{\prime \prime }}\right| ^{2}\left( 2C_{G,l^{\prime \prime }}\right) ^{r}\left( {{\mathsf {E}}}N_{1}^{2r}\right) C_{B}h_{ {\mathrm{s}}}^{1-r}+\mathcal {U}(1)\right] \\&\le 2^{r}\left( cD_{n}\right) ^{r-2}\left( 2C_{G,l^{\prime \prime }}\right) ^{r}c_{N}^{r}r!C_{B}h_{{\mathrm{s}}}^{2-r}c_{\zeta }^{-1} {{\mathsf {E}}}\left( \zeta _{i,J,2}^{*}\right) ^{2} \\&\le \left( C_{\zeta }D_{n}h_{{\mathrm{s}}}^{-1}\right) ^{r-2}r! {{\mathsf {E}}}\left( \zeta _{i,J,2}^{*}\right) ^{2}, \end{aligned}$$

which implies that $\left\{ \zeta _{i,J,2}^{*}\right\} _{i=1}^{n}$ satisfies Cramér’s condition. Applying Lemma 1 to $\sum _{i=1}^{n}\zeta _{i,J,2}^{*}$, for $r>2$ and any large enough $\delta >0$,

$$\begin{aligned}&P\left\{ \left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,2}^{*}\right| \ge \delta (nh_{{\mathrm{s}}})^{-1/2}(\log (n))^{1/2}\right\} \\&\quad \le 2\exp \left\{ \frac{-\delta ^{2}\log (n)}{4+2C_{\zeta }D_{n}h_{{\mathrm{s}}}^{-1}\delta \left( \log (n)\right) ^{1/2}n^{-1/2}h_{{\mathrm{s }}}^{1/2}}\right\} \le 2n^{-8}. \end{aligned}$$

Hence

$$\begin{aligned} \sum _{n=1}^{\infty }P\left\{ \sup _{J,l^{\prime \prime },l}\left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,2}^{*}\right| \ge \delta \left( nh_{ {\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} <\infty . \end{aligned}$$

Thus, $\sup _{J,l^{\prime \prime },l}\left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,2}^{*}\right| =\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}} }\right) ^{-1/2}(\log (n))^{1/2}\right\}$ as $n\rightarrow \infty$ by the Borel–Cantelli lemma. Furthermore, we have

$$\begin{aligned} A_{n,\xi }&\le \sup _{J,l,l^{\prime \prime }}\left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,1}\right| +\sup _{J,l^{\prime \prime },l}\left| n^{-1}\sum _{i=1}^{n}\zeta _{i,J,2}^{*}\right| +\sup _{J,l^{\prime \prime },l}\left| {{\mathsf {E}}}\left( \zeta _{i,J,1}\right) \right| \\&= \mathcal {U}_{a.s.}\left( n^{-k}\right) +\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} +\mathcal {U}\left( D_{n}^{-\left( 1+\eta /4\right) }\right) \\&= \mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

The properties of $R_{ij,\varepsilon ,J,l}$ are obtained similarly. $\square$

Next define two $d\times d$ matrices

$$\begin{aligned}&\mathbf {\Gamma }_{\xi ,n}(t) =c_{J(t),n}^{-1}N_{{\mathrm{T}} }^{-2}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }\left\{ \sum _{j=1}^{N_{i}}B_{J(t)}(T_{ij})\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} ^{2}X_{il^{\prime \prime }}^{2}\mathbf {X}_{i} \mathbf {X}_{i}^{\scriptstyle {\mathsf {T}}}, \\&\mathbf {\Gamma }_{\varepsilon ,n}(t) =c_{J(t),n}^{-1}N_{{\mathrm{T}} }^{-2}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J(t)}^{2}(T_{ij})\sigma ^{2}\left( T_{ij}\right) \mathbf {X}_{i}\mathbf {X}_{i}^{\scriptstyle {\mathsf {T}}}. \end{aligned}$$

Lemma 6

For any $t\in \mathbb {R}$, the conditional covariance matrices of $\hat{\varvec{\xi }}\left( t\right)$ and $\hat{\varvec{ \varepsilon }}\left( t\right)$ on $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$ are

$$\begin{aligned}&\mathbf {\Sigma }_{\xi ,n}(t)={{\mathsf {E}}}\left\{ \hat{\varvec{\xi }}\left( t\right) \hat{\varvec{\xi }}^{\scriptstyle {\mathsf {T}}}\left( t\right) \left| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right. \right\} = \mathbf {H}^{-1}\mathbf {\Gamma }_{\xi ,n}(t)\mathbf {H}^{-1},\\&\mathbf {\Sigma }_{\varepsilon ,n}(t)={{\mathsf {E}}}\left\{ \hat{\varvec{\varepsilon }} \left( t\right) \hat{\varvec{\varepsilon }}^{\scriptstyle {\mathsf {T}}}\left( t\right) \left| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right. \right\} =\mathbf {H}^{-1}\mathbf {\Gamma } _{\varepsilon ,n}(t)\mathbf {H}^{-1}, \end{aligned}$$

and with $\mathbf {\Sigma }_{n}(t)$ defined in (7),

$$\begin{aligned} \sup _{t\in [0,1]}\left\| \left\{ \mathbf {\Sigma }_{\xi ,n}(t)+ \mathbf {\Sigma }_{\varepsilon ,n}(t)\right\} -\mathbf {\Sigma } _{n}(t)\right\| _{\infty }=\mathcal {O}_{a.s.}\left\{ n^{-3/2}h_{{\mathrm{s}}}^{-3/2}( \log (n))^{1/2}\right\} . \end{aligned}$$

(33)

Proof

Note that

$$\begin{aligned} \hat{\varvec{\xi }}\left( t\right) \hat{\varvec{\xi }}^{\scriptstyle { \mathsf {T}}}\left( t\right)&= c_{J(t),n}^{-1}\mathbf {H}^{-1}\left\{ \frac{1 }{N_{{\mathrm{T}}}^{2}}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J(t)}(T_{ij})X_{il} \sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\xi _{ik,l^{\prime \prime }}\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) X_{il^{\prime \prime }}\right. \\&\quad \times \left. \sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J(t)}(T_{ij})X_{il^{\prime }}\sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\xi _{ik,l^{\prime \prime }}\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) X_{il^{\prime \prime }}\right\} _{l,l^{\prime }=1}^{d}\mathbf {H}^{-1}. \end{aligned}$$

Thus,

$$\begin{aligned} \mathbf {\Sigma }_{\xi ,n}(t)&= {{\mathsf {E}}}\left\{ \hat{ \varvec{\xi }}\left( t\right) \hat{\varvec{\xi }}^{\scriptstyle { \mathsf {T}}}\left( t\right) \left| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right. \right\} =c_{J(t),n}^{-1}\mathbf {H}^{-1} \\&\times \left[ N_{{\mathrm{T}}}^{-2}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }\left\{ \sum _{j=1}^{N_{i}}B_{J(t)}(T_{ij})\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} ^{2}X_{il^{\prime \prime }}^{2}\mathbf {X}_{i}\mathbf {X }_{i}^{\scriptstyle {\mathsf {T}}}\right] \mathbf {H}^{-1} \\ \qquad \qquad&= \mathbf {H}^{-1}\mathbf {\Gamma }_{\xi ,n}(t)\mathbf {H}^{-1}. \end{aligned}$$

Similarly, we can derive the conditional covariance matrix of $\hat{ \varvec{\varepsilon }}\left( t\right)$. Next let

$$\begin{aligned} \Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}&= \left\{ \sum _{j=1}^{N_{i}}B_{J}(T_{ij})\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) \right\} ^{2}X_{il^{\prime \prime }}^{2}X_{il}X_{il^{\prime }},\\ \Psi _{ij,\varepsilon ,J,l,l^{\prime }}&= B_{J}^{2}(T_{ij})\sigma ^{2}\left( T_{ij}\right) X_{il}X_{il^{\prime }}. \end{aligned}$$

Similar to the proof of Lemma 5,

$$\begin{aligned} {{\mathsf {E}}}\left( \sum _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}\right)&= c_{J,n}^{-1}{{\mathsf {E}}}\left( X_{il^{\prime \prime }}^{2}X_{il}X_{il^{\prime }}\right) \left[ \left( {{\mathsf {E}}}N_{1}\right) \int \nolimits _{\chi _{J}}G_{l^{\prime \prime }}\left( u,u\right) f\left( u\right) \mathrm{d}u\right. \\&\left. +{{\mathsf {E}}}\left\{ N_{1}(N_{1}-1)\right\} \int \nolimits _{\chi _{J}\times \chi _{J}}G_{l^{\prime \prime }}\left( u,v\right) f\left( u\right) f\left( v\right) \mathrm{d}u\mathrm{d}v\right] ,\\ {{\mathsf {E}}}\Psi _{ij,\varepsilon ,J,l,l^{\prime }}&= c_{J,n}^{-1} {{\mathsf {E}}}\left( X_{il}X_{il^{\prime }}\right) \int \nolimits _{\chi _{J}}\sigma ^{2}\left( u\right) f\left( u\right) \mathrm{d}u, \end{aligned}$$

and as $n\rightarrow \infty$,

$$\begin{aligned}&\sup _{J,l,l^{\prime },l^{\prime \prime }}\left| n^{-1}\sum _{i=1}^{n}\sum _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}-{{\mathsf {E}}}\left( \sum _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}\right) \right| \\&\quad = \mathcal {O }_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} , \end{aligned}$$

$$\begin{aligned} \sup _{J,l,l^{\prime }}\left| N_{{\mathrm{T}}}^{-1}\sum _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}\Psi _{ij,\varepsilon ,J,l,l^{\prime }}- {{\mathsf {E}}}\Psi _{ij,\varepsilon ,J,l,l^{\prime }}\right| = \mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

Furthermore,

$$\begin{aligned}&\sup _{J,l,l^{\prime },l^{\prime \prime }}\left| N_{T}^{-2}\sum \limits _{i=1}^{n}\sum \limits _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}-n^{-1}\left( {{\mathsf {E}}} N_{1}\right) ^{-2}{{\mathsf {E}}}\left( \sum \limits _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}\right) \right| \\&\qquad \le \sup _{J,l,l^{\prime },l^{\prime \prime }}n^{-1}\left( {{\mathsf {E}}}N_{1}\right) ^{-2}\left\{ \left| \left( \frac{n {{\mathsf {E}}}N_{1}}{N_{T}}\right) ^{2}-1\right| \left| n^{-1}\sum \limits _{i=1}^{n}\sum \limits _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}\right| \right. \\&\qquad \quad \left. +\left| n^{-1}\sum \limits _{i=1}^{n}\sum \limits _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}-{{\mathsf {E}}} \left( \sum \limits _{k=1}^{\infty }\Psi _{ik,\xi ,J,l,l^{\prime },l^{\prime \prime }}\right) \right| \right\} \\&\qquad =\mathcal {O}_{a.s.}\left\{ n^{-3/2}h_{{\mathrm{s}}}^{-1/2}(\log (n))^{1/2}\right\} , \end{aligned}$$

and

$$\begin{aligned}&\sup _{J,l,l^{\prime }}\left| N_{T}^{-2}\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}\Psi _{ik,\varepsilon ,J,l,l^{\prime }}-\left( n{{\mathsf {E}}}N_{1}\right) ^{-1}{{\mathsf {E}}}\Psi _{ik,\varepsilon ,J,l,l^{\prime }}\right| \\&\qquad \le \sup _{J,l,l^{\prime }}\left( n{{\mathsf {E}}}N_{1}\right) ^{-1}\left\{ \left| \frac{n{{\mathsf {E}}}N_{1}}{N_{T}} -1\right| \left| N_{T}^{-1}\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}\Psi _{ik,\varepsilon ,J,l,l^{\prime }}\right| \right. \\&\qquad \quad \left. +\left| N_{T}^{-1}\sum \limits _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}\Psi _{ik,\varepsilon ,J,l,l^{\prime }}-{{\mathsf {E}}}\Psi _{ik,\varepsilon ,J,l,l^{\prime }}\right| \right\} \\&\qquad =\mathcal {O}_{a.s.}\left\{ n^{-3/2}h_{{\mathrm{s}}}^{-1/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

Notice that

$$\begin{aligned}&\mathbf {\Sigma }_{n}(t)=\mathbf {H}^{-1}c_{J(t),n}^{-1}\left( n {{\mathsf {E}}}N_{1}\right) ^{-1}\\&\qquad \qquad \times \left\{ \left( {{\mathsf {E}}} N_{1}\right) ^{-1}{{\mathsf {E}}}\left( \sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\Psi _{ik,\xi ,J\left( t\right) ,l,l^{\prime },l^{\prime \prime }}\right) +{{\mathsf {E}}}\Psi _{ij,\varepsilon ,J\left( t\right) ,l,l^{\prime }}\right\} _{l,l^{\prime }=1}^{d}\mathbf {H}^{-1}, \\&\mathbf {\Sigma }_{\xi ,n}(t)+\mathbf {\Sigma }_{\varepsilon ,n}(t) \\&\quad =\mathbf {H}^{-1}c_{J(t),n}^{-1}N_{{\mathrm{T}}}^{-2}\left\{ \sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }\Psi _{ik,\xi ,J(t),l,l^{\prime },l^{\prime \prime }}\!+\!\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}\Psi _{ij,\varepsilon ,J(t),l,l^{\prime }}\right\} _{l,l^{\prime }\!=\!1}^{d}\mathbf {H}^{-1}, \end{aligned}$$

and (35) implies $\sup _{t\in \left[ 0,1\right] }\left| c_{J(t),n}\right| =\mathcal {O}\left( h_{{\mathrm{s}}}\right)$. Hence (33) holds. $\square$

Given $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$, let $\sigma _{\xi _{l},n}^{2}(t)$ and $\sigma _{\varepsilon _{l},n}^{2}(t)$ be the conditional variances of $\hat{\xi }_{l}(t)$ and $\hat{\varepsilon } _{l}(t)$ defined in (38) and (39), respectively. Lemma 6 implies that

$$\begin{aligned} \sup _{t\in [0,1]}\left| \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)-\sigma _{n,ll}^{2}(t)\right| =\mathcal {O} _{a.s.}\left\{ n^{-3/2}h_{{\mathrm{s}}}^{-3/2}(\log (n))^{1/2}\right\} . \end{aligned}$$

(34)

Lemma 7

Under Assumptions (A2)–(A6), for $l=1,\ldots , d$, $\eta _{l}(t)$ defined in (40) is a Gaussian process consisting of $\left( N_{{\mathrm{s}} }+1\right)$ standard normal variables $\left\{ \eta _{J,l}\right\} _{J=0}^{N_{{\mathrm{s}}}}$ such that $\eta _{l}(t)=\eta _{J(t),l}$ for $t\in \left[ 0,1\right]$, and there exists a constant $C>0$ such that for large $n$, $\sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| {{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}\right| \le Ch_{{\mathrm{s}}}$.

Proof

For any fixed $l=1,\ldots ,d$ and $0\le J\le N_{{\mathrm{s}}}$, $\mathcal {L}\left\{ \eta _{J,l}\left| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right. \right\} =N\left( 0,1\right)$ by Assumption (A2), so $\mathcal {L}\left\{ \eta _{J,l}\right\} =N\left( 0,1\right)$, for $0\le J\le$ $N_{{\mathrm{s}}}$.

Next we derive the upper bound for $\sup _{0\le J\ne J^{\prime }\le N_{ {\mathrm{s}}}}\left| {{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}\right|$. Let

$$\begin{aligned} \bar{R}_{\xi ,J(t),l}=N_{{\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J(t),l^{\prime \prime },l}^{2},\quad \bar{R}_{\varepsilon ,J(t),l}=N_{{\mathrm{T}} }^{-1}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}R_{ij,\varepsilon ,J(t),l}^{2}, \end{aligned}$$

then we have

$$\begin{aligned} \sigma _{\xi _{l},n}(t)&= \left\{ c_{J(t),n}^{-1}N_{{\mathrm{T}} }^{-2}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J(t),l^{\prime \prime },l}^{2}\right\} ^{1/2}=\left\{ c_{J(t),n}^{-1}N_{{\mathrm{T}}}^{-1}\bar{R}_{\xi ,J(t),l}\right\} ^{1/2},\\ \sigma _{\varepsilon _{l},n}(t)&= \left\{ c_{J(t),n}^{-1}N_{{\mathrm{T}} }^{-2}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}R_{ij,\varepsilon ,J(t),l}^{2}\right\} ^{1/2}=\left\{ c_{J(t),n}^{-1}N_{{\mathrm{T}}}^{-1}\bar{R}_{ \varepsilon ,J(t),l}\right\} ^{1/2}. \end{aligned}$$

For $J\ne J^{\prime }$, by (31) and the definition of $B_{J}$,

$$\begin{aligned} R_{ij,\varepsilon ,J,l}R_{ij,\varepsilon ,J^{\prime },l}=\left( \sum _{l^{\prime }=1}^{d}z_{ll^{\prime }}X_{il^{\prime }}\right) ^{2}B_{J}\left( T_{ij}\right) B_{J^{\prime }}\left( T_{ij}\right) \sigma ^{2}\left( T_{ij}\right) =0, \end{aligned}$$

along with the conditional independence of $\hat{\xi }_{l}(t),\hat{\varepsilon }_{l}(t)$ on $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$, and independence of $\xi _{ik,l}$, $T_{ij},N_{i}$, $\left\{ X_{il}\right\} _{l=1}^{d}$, $1\le j\le N_{i}$, $1\le i\le n$, $k=1,2,\ldots$,

$$\begin{aligned}&{{\mathsf {E}}}\left( \eta _{J,l}\eta _{J^{\prime },l}\right) = {{\mathsf {E}}}\left[ (\bar{R}_{\xi ,J,l}+\bar{R}_{ \varepsilon ,J,l})^{-1/2}(\bar{R}_{\xi ,J^{\prime },l}+\bar{R} _{\varepsilon ,J^{\prime },l})^{-1/2}\right. \\&\qquad \times N_{{\mathrm{T}}}^{-1}{{\mathsf {E}}}\left\{ \left( \sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}\xi _{ik,l^{\prime \prime }}\right) \left( \sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n} \sum _{k=1}^{\infty }R_{ik,\xi ,J^{\prime },l^{\prime \prime },l}\xi _{ik,l^{\prime \prime }}\right) \right. \\&\qquad \left. \left. \left. +\left( \sum _{i=1}^{n}\sum _{j=1}^{N_{i}}R_{ij,\varepsilon ,J,l}\varepsilon _{ij}\right) \left( \sum _{i=1}^{n}\sum _{j=1}^{N_{i}}R_{ij,\varepsilon ,J^{\prime },l}\varepsilon _{ij}\right) \right| \left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}\right\} \right] \\&\quad ={{\mathsf {E}}}C_{n,J,J^{\prime },l}, \end{aligned}$$

in which

$$\begin{aligned} C_{n,J,J^{\prime },l}&= (\bar{R}_{\xi ,J,l}+\bar{R}_{\varepsilon ,J,l})^{-1/2}( \bar{R}_{\xi ,J^{\prime },l}+\bar{R}_{\varepsilon ,J^{\prime },l})^{-1/2}\\&\times \left\{ N_{{\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}R_{ik,\xi ,J^{\prime },l^{\prime \prime },l}\right\} . \end{aligned}$$

Note that according to definitions of $R_{ik,\xi ,J,l^{\prime \prime },l},R_{ij,\varepsilon ,J,l}$, and Lemma 5, for $0\le J\le N_{{\mathrm{s}}}$

$$\begin{aligned}&\displaystyle \bar{R}_{\xi ,J(t),l}+\bar{R}_{\varepsilon ,J(t),l}\ge \bar{R} _{\varepsilon ,J(t),l}\ge ER_{ij,\varepsilon ,J,l}^{2}-A_{n,\varepsilon }\ge c_{R}-A_{n,\varepsilon },\\&\displaystyle P\left[ \,\inf _{0 \le J \ne J^{\prime } \le N_{{\mathrm{s}}}}\left\{ (\bar{R}_{\xi ,J,l}\!+\!\bar{R}_{\varepsilon ,J,l})(\bar{R}_{\xi ,J^{\prime },l}\!+\!\bar{R} _{\varepsilon ,J^{\prime },l})\right\} \! \!\ge \! \left( c_{R}-\delta \sqrt{\frac{ \log (n)}{nh_{{\mathrm{s}}}}}\right) ^{2}\right] \!\!\ge \! 1\!-\!2n^{-8}. \end{aligned}$$

Thus for large $n$, with probability $\ge 1-2n^{-8}$, the denominator of $C_{n,J,J^{\prime },l}$ is uniformly greater than $c_{R}^{2}/4$. On the other hand, we consider the numerator of $C_{n,J,J^{\prime },l}$.

$$\begin{aligned}&{{\mathsf {E}}}\left( N_{{\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }\!=\!1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}R_{ik,\xi ,J^{\prime },l^{\prime \prime },l}\right) \!=\! {{\mathsf {E}}}\left\{ N_{{\mathrm{T}}}^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\left( \sum _{l^{\prime }\!=\!1}^{d}z_{ll^{\prime }}X_{il^{\prime }}X_{il^{\prime \prime }}\right) ^{2}\right. \\&\quad \times \left. \! \left( \sum _{j=1}^{N_{i}}\sum _{j^{\prime }=1}^{N_{i}}B_{J}\left( T_{ij}\right) B_{J^{\prime }}\left( T_{ij^{\prime }}\right) G_{l^{\prime \prime }}\left( T_{ij},T_{ij^{\prime }}\right) \!\right) \! \right\} \!\sim h_{{\mathrm{s}}}. \end{aligned}$$

Applying Bernstein’s inequality, there exists $C_{0}>0$ such that, for large $n$,

$$\begin{aligned} P\left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| N_{{\mathrm{T}} }^{-1}\sum _{l^{\prime \prime }=1}^{d}\sum _{i=1}^{n}\sum _{k=1}^{\infty }R_{ik,\xi ,J,l^{\prime \prime },l}R_{ik,\xi ,J^{\prime },l^{\prime \prime },l}\right| \le C_{0}h_{{\mathrm{s}}}\right) \ge 1-2n^{-8}. \end{aligned}$$

Putting the above together, for large $n$, $C_{1}=C_{0}\left( c_{R}^{2}/4\right) ^{-1}$,

$$\begin{aligned} P\left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| C_{n,J,J^{\prime },l}\right| \le C_{1}h_{{\mathrm{s}}}\right) \ge 1-4n^{-8}. \end{aligned}$$

Note that as a continuous random variable, $\sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| C_{n,J,J^{\prime },l}\right| \in \left[ 0,1 \right] \,$, thus

$$\begin{aligned} {{\mathsf {E}}}\left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}} }}\left| C_{n,J,J^{\prime },l}\right| \right) =\int \nolimits _{0}^{1}P\left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| C_{n,J,J^{\prime },l}\right| >u\right) \mathrm{d}u. \end{aligned}$$

For large $n$, $C_{1}h_{{\mathrm{s}}}<1$ and then ${{\mathsf {E}}}\left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}},l}\left| C_{n,J,J^{\prime }}\right| \right)$ is

$$\begin{aligned}&\int \nolimits _{0}^{C_{1}h_{{\mathrm{s}}}}P\left\{ \,\sup _{0\le J\ne J^{\prime }\le N_{ {\mathrm{s}}},l}\left| C_{n,J,J^{\prime },l}\right| \!>\!u\right\} \mathrm{d}u\!+\!\int \nolimits _{C_{1}h_{{\mathrm{s}}}}^{1}P\left\{ \sup _{0\le J \ne J^{\prime }\le N_{ {\mathrm{s}}},l}\left| C_{n,J,J^{\prime },l}\right| >u\right\} \mathrm{d}u \end{aligned}$$

$$\begin{aligned}&\le \int \nolimits _{0}^{C_{1}h_{{\mathrm{s}}}}1\mathrm{d}u+\int \nolimits _{C_{1}h_{{\mathrm{s}} }}^{1}4n^{-8}\mathrm{d}u\le C_{1}h_{{\mathrm{s}}}+4n^{-8}\le Ch_{{\mathrm{s}}} \end{aligned}$$

for some $C>0$ and large enough $n$. The lemma now follows from

$$\begin{aligned} \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| {{\mathsf {E}}}\left( C_{n,J,J^{\prime },l}\right) \right| \le {{\mathsf {E}}} \left( \sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}}\left| C_{n,J,J^{\prime },l}\right| \right) \le Ch_{{\mathrm{s}}}. \end{aligned}$$

This completes the proof of the lemma. $\square$

Lemma 8

Under Assumptions (A2)–(A6), for $\eta _{l}(t),\sigma _{n,ll}(t),l=1,\ldots , d$, defined in (40) and (7), one has $\left| \sigma _{n,ll}(t)^{-1}\left\{ \hat{\xi }_{l}(t)+ \hat{\varepsilon }_{l}(t)\right\} -\eta _{l}(t)\right| =\left| r_{n,l}(t)-1\right| \left| \eta _{l}(t)\right|$, where $r_{n,l}(t)=\sigma _{n,ll}^{-1}(t)\left\{ \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)\right\} ^{1/2}$, and as $n\rightarrow \infty$,

$$\begin{aligned} \sup _{t\in \left[ 0,1\right] }\left\{ a_{N_{{\mathrm{s}}}+1}\left| r_{n,l}(t)-1\right| \right\} =\mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}} }\right) ^{-1/2}(\log \left( N_{{\mathrm{s}}}+1\right) \log (n))^{1/2}\right\} . \end{aligned}$$

Proof

By Lemma 5, $\sigma _{n,ll}^{2}(t)$ in (7) can be rewritten as

$$\begin{aligned}&\sigma _{n,ll}^{2}(t) =c_{J(t),n}^{-1}\left( n{{\mathsf {E}}} N_{1}\right) ^{-1}\left\{ \left( {{\mathsf {E}}}N_{1}\right) ^{-1}\sum _{l^{\prime \prime }=1}^{d}{{\mathsf {E}}}\left( \sum _{k=1}^{\infty }R_{ik,\xi ,J(t),l^{\prime \prime },l}^{2}\right) + {{\mathsf {E}}}R_{ij,\varepsilon ,J(t),l}^{2}\right\} \\&\qquad \quad \!\! \sim n^{-1}h_{{\mathrm{s}}}^{-1}. \end{aligned}$$

Hence, according to (34) and (10),

$$\begin{aligned} \sup _{t\in \left[ 0,1\right] }\left\{ a_{N_{{\mathrm{s}}}+1} \left| r_{n,l}(t)-1\right| \right\}&= \sup _{t\in \left[ 0,1\right] }\left\{ a_{N_{{\mathrm{s}}}+1}\left| \sigma _{n,ll}^{-1}(t) \left\{ \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)\right\} ^{1/2}-1\right| \right\} \\&\le \sup _{t\in \left[ 0,1\right] }\left\{ a_{N_{{\mathrm{s}}}+1} \left| \sigma _{n,ll}^{-2}(t)\left\{ \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)\right\} -1\right| \right\} \\&= \sup _{t\in \left[ 0,1\right] }\left\{ a_{N_{{\mathrm{s}}}+1} \sigma _{n,ll}^{-2}(t)\left| \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)-\sigma _{n,ll}^{2}(t)\right| \right\} \\&= \mathcal {O}_{a.s.}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{-1/2}( \log \left( N_{{\mathrm{s}}}+1\right) \log (n))^{1/2}\right\} . \end{aligned}$$

This completes the proof. $\square$

1.3 A.3 Proofs of Propositions 1– 4

Proof of Proposition 1

By Assumption (A3) on the continuity of functions $\phi _{k,l}(t)$, $\sigma ^{2}(t)$ and $f(t)$ on $\left[ 0,1\right]$ and Assumption (A4), for any $t,u\in \left[ 0,1\right]$ satisfying $\left| t-u\right| \le h_{{\mathrm{s}}}$,

$$\begin{aligned} \left| G_{l}(t,t)-G_{l}(u,u)\right| \le \sum _{k=1}^{\infty }\left| \phi _{k,l}^{2}(t)-\phi _{k,l}^{2}(u)\right| \le 2\sum _{k=1}^{\infty }\left\| \phi _{k,l}\right\| _{\infty } \omega \left( \phi _{k,l},h_{{\mathrm{s}}}\right) \le Ch_{{\mathrm{s}}}^{r}. \end{aligned}$$

Furthermore,

$$\begin{aligned} \left| \int \nolimits _{{\chi _{J(t)}}}\left\{ G_{l}(t,t)f(t)-G_{l}(u,u)f\left( u\right) \right\} \mathrm{d}u\right| \le Ch_{{\mathrm{s}}}^{1+r}=\mathcal {O}\left( h_{{\mathrm{s}}}^{1+r}\right) , \end{aligned}$$

$$\begin{aligned} \left| \int \nolimits _{\chi _{J(t)}\times \chi _{J(t)}}\left\{ G_{l}(t,t)f^{2}(t)-G_{l}\left( u,v\right) f\left( u\right) f\left( v\right) \right\} \mathrm{d}u\mathrm{d}v\right| \le Ch_{{\mathrm{s}}}^{2+r}=\mathcal {O}\left( h_{{\mathrm{ s}}}^{2+r}\right) , \end{aligned}$$

$$\begin{aligned} \left| \int \nolimits _{\chi _{J(t)}}\left\{ \sigma ^{2}(t) f(t)-\sigma ^{2}\left( u\right) f\left( u\right) \right\} \mathrm{d}u\right| \le Ch_{{\mathrm{s}}}^{1+r}= \mathcal {O}\left( h_{{\mathrm{s}}}^{1+r}\right) . \end{aligned}$$

According to the definition of $C_{J,n}$ in (6),

$$\begin{aligned} C_{J,n}=\int \nolimits _{[\upsilon _{J},\upsilon _{J+1}]}f(x)\mathrm{d}x=f(\upsilon _{J})h_{{\mathrm{s}} }+\int \nolimits _{[\upsilon _{J},\upsilon _{J+1}]} \{f(x)-f(\upsilon _{J})\}\mathrm{d}x, \end{aligned}$$

(35)

thus, $|C_{J,n}-f(\upsilon _{J})h_{{\mathrm{s}}}|\le w(f,h_{{\mathrm{s}}})h_{{\mathrm{s}} }$ for all $J=0,\ldots ,N_{{\mathrm{s}}}$. Therefore,

$$\begin{aligned} \mathbf {\Gamma }_{n}(t)&= \left\{ f(t)h_{{\mathrm{s}}}+\mathcal {U}\left( h_{ {\mathrm{s}}}^{1+r}\right) \right\} ^{-2}\left( n{{\mathsf {E}}} N_{1}\right) ^{-1}{{\mathsf {E}}}\left[ \left\{ \sigma _{Y}^{2}\left( t,\mathbf {X}\right) f\left( t\right) h_{{\mathrm{s}}}+\mathcal {U}_{p}\left( h_{ {\mathrm{s}}}^{1+r}\right) \right. \right. \\&\left. \left. \quad +\frac{{{\mathsf {E}}}\left\{ N_{1}(N_{1}-1)\right\} }{ {{\mathsf {E}}}N_{1}}\sum _{l=1}^{d}X_{l}^{2}G_{l}\left( t,t\right) f^{2}\left( t\right) h_{{\mathrm{s}}}^{2}+\mathcal {U}_{p}\left( h_{{\mathrm{s}} }^{2+r}\right) \right\} \mathbf {XX}^{\scriptstyle {\mathsf {T}}}\right] \\&= {{\mathsf {E}}}\left[ \mathbf {XX}^{\scriptstyle {\mathsf {T}}}\sigma _{Y}^{2}\left( t,\mathbf {X}\right) \left\{ f(t)h_{{\mathrm{s}}}n {{\mathsf {E}}}N_{1}\right\} ^{-1}\left\{ 1+\frac{{{\mathsf {E}}} \left\{ N_{1}(N_{1}-1)\right\} }{{{\mathsf {E}}}N_{1}}\right. \right. \\&\quad \left. \left. \times \frac{\sum _{l=1}^{d}X_{l}^{2}G_{l}\left( t,t\right) f\left( t\right) h_{{\mathrm{s}}}}{\sigma _{Y}^{2}\left( t,\mathbf {X}\right) } \right\} \left\{ 1+\mathcal {U}_{p}\left( h_{{\mathrm{s}}}^{r}\right) \right\} \right] =\tilde{\mathbf {\Gamma }}_{n}(t)+\mathcal {U}\left( n^{-1}h_{{\mathrm{s}} }^{r-1}\right) , \end{aligned}$$

establishing the proposition. $\square$

Proof of Proposition 2

The result follows from standard theory of kernel and spline smoothing, as in Wang and Yang (2009), thus omitted. $\square$

Proof of Proposition 3

According to the result on page 149 of de Boor (2001), there exist functions $g_{l}\in G^{(-1)}\left[ 0,1\right]$ that satisfies $\left\| m_{l}-g_{l}\right\| _{\infty }=\mathcal {O}\left( h_{{\mathrm{s}}}\right)$ for $l=1,\ldots ,d$. By the definition of $\tilde{m}_{l}\left( t\right)$ in ( 22),

$$\begin{aligned} \tilde{\varvec{m}}(t)=\left( \tilde{m}_{1}(t),\ldots ,\tilde{m} _{d}(t)\right) ^{\scriptstyle {\mathsf {T}}}=c_{J(t),n}^{-1/2}\left( \tilde{ \gamma }_{J(t),1},\ldots ,\tilde{\gamma }_{J(t),d}\right) ^{\scriptstyle { \mathsf {T}}}=c_{J(t),n}^{-1/2}\tilde{\mathbf {\gamma }}_{J(t)}, \end{aligned}$$

where $\tilde{\mathbf {\gamma }}_{J}=\hat{\mathbf {{V}}}_{J}^{-1}\left\{ N_{ {\mathrm{T}}}^{-1}\sum \nolimits _{i=1}^{n}\sum \nolimits _{j=1}^{N_{i}}B_{J}(T_{ij})X_{il}\sum \nolimits _{l^{\prime }=1}^{d}m_{l^{\prime }}(T_{ij})X_{il^{\prime }}\right\} _{l=1}^{d}$ for $\hat{\mathbf {{V}}}_{J}$ defined in (18).

Let $\tilde{\varvec{g}}(t)=(\tilde{g}_{1}\left( t\right) ,\ldots ,\tilde{g} _{d}\left( t\right) )^{\scriptstyle {\mathsf {T}}}$, then $\tilde{\mathbf {m}} _{l}\left( t\right) -\tilde{\mathbf {g}}_{l}\left( t\right)$ equals to

$$\begin{aligned} c_{J(t),n}^{-1/2}\hat{\mathbf {V}}_{J(t)}^{-1}\left[ \frac{1}{N_{{\mathrm{T}}}} \sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J(t)}(T_{ij})X_{il}\sum \limits _{l^{\prime }=1}^{d}\left\{ m_{l^{\prime }}(T_{ij})-g_{l^{\prime }}(T_{ij})\right\} X_{il^{\prime }}\right] _{l=1}^{d}. \end{aligned}$$

Observing that $\tilde{g}_{l}\equiv g_{l}$ as $g_{l}\in G^{(-1)}\left[ 0,1 \right]$, there is a decomposition similar to (24), $\tilde{m}_{l}\left( t\right) =\tilde{m}_{l}\left( t\right) -\tilde{g} _{l}\left( t\right) +g_{l}\left( t\right)$, $l=1,\ldots ,d$.

By (35), $\sup _{t\in \left[ 0,1\right] }\left| c_{J(t),n}\right| =\mathcal {O}\left( h_{{\mathrm{s}}}\right)$. Next ${{\mathsf {E}}}|B_{J}(T_{ij})|=c_{J,n}^{-1/2}\int b_{J}(x)f(x)\mathrm{d}x \sim h_{{\mathrm{s}}}^{1/2}$, thus $\sup _{t\in \left[ 0,1\right] }\left| B_{J(t)}(T_{ij})\right| =\mathcal {O}_{p}(h_{{\mathrm{s}}}^{1/2})$. Then it is easy to show that $\left\| \tilde{m}_{l}-\tilde{g}_{l}\right\| _{\infty }=\mathcal {O}_{p}\left( h_{{\mathrm{s}}}^{-1/2}h_{{\mathrm{s}}}^{1/2}h_{ {\mathrm{s}}}\right) =\mathcal {O}_{p}\left( h_{{\mathrm{s}}}\right)$. Hence, for $l=1,\ldots ,d$,

$$\begin{aligned} \left\| \tilde{m}_{l}-m_{l}\right\| _{\infty }\le \left\| \tilde{m} _{l}-\tilde{g}_{l}\right\| _{\infty }+\left\| m_{l}-g_{l}\right\| _{\infty }=\mathcal {O}_{p}\left( h_{{\mathrm{s}}}\right) , \end{aligned}$$

which completes the proof. $\square$

Note that $B_{J}(t)\equiv b_{J}c_{J,n}^{-1/2}$, $t\in \left[ 0,1\right]$, so the terms $\tilde{\xi }_{l}(t)$ and $\tilde{\varepsilon }_{l}(t)$, $l=1,\ldots ,d$, defined in (23) are

$$\begin{aligned} \tilde{\varvec{\xi }}(t)&= \left( \tilde{\xi }_{1}(t),\ldots ,\tilde{\xi } _{d}(t)\right) ^{\scriptstyle {\mathsf {T}}}=c_{J(t),n}^{-1/2}\left( \tilde{ \alpha }_{J(t),1},\ldots ,\tilde{\alpha }_{J(t),d}\right) ^{\scriptstyle { \mathsf {T}}}=c_{J(t),n}^{-1/2}\tilde{\mathbf {\varvec{\alpha } }}_{J(t)}, \quad \end{aligned}$$

(36)

$$\begin{aligned} \tilde{\varvec{\varepsilon }}(t)&= \left( \tilde{\varepsilon } _{1}(t),\ldots ,\tilde{\varepsilon }_{d}(t)\right) ^{\scriptstyle {\mathsf {T}} }=c_{J(t),n}^{-1/2}\left( \tilde{\theta }_{J(t),1},\ldots ,\tilde{\theta } _{J(t),d}\right) ^{\scriptstyle {\mathsf {T}}}=c_{J(t),n}^{-1/2}\tilde{\mathbf { \varvec{\theta } }}_{J(t)}, \end{aligned}$$

(37)

where

$$\begin{aligned} \tilde{\mathbf {\alpha }}_{J}&= \hat{\mathbf {{V}}}_{J}^{-1}\left\{ N_{{\mathrm{T}} }^{-1}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J}(T_{ij})X_{il}\sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\xi _{ik,l^{\prime \prime }}\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) X_{il^{\prime \prime }}\right\} _{l=1}^{d}, \\ \tilde{\mathbf {\theta }}_{J}&= \hat{\mathbf {{V}}}_{J}^{-1}\left\{ N_{{\mathrm{T}} }^{-1}\sum _{i=1}^{n}\sum _{j=1}^{N_{i}}B_{J}(T_{ij})X_{il}\sigma \left( T_{ij}\right) \varepsilon _{ij}\right\} _{l=1}^{d}. \end{aligned}$$

According to Lemma 3, the inverse of the random matrix $\hat{\mathbf {V}}_{J}$ can be approximated by that of a deterministic matrix $\mathbf {H}={{\mathsf {E}}}(\mathbf {XX}^{\scriptstyle {\mathsf {T}}})$. Substituting $\hat{\mathbf {V}}_{J}$ with $\mathbf {H}$ in (36) and (37), we define the random vectors

$$\begin{aligned} \hat{\varvec{\xi }}(t)&= c_{J(t),n}^{-1/2}\mathbf {H}^{-1}\left\{ \frac{1 }{N_{{\mathrm{T}}}}\sum _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}B_{J(t)}(T_{ij})X_{il} \sum _{l^{\prime \prime }=1}^{d}\sum _{k=1}^{\infty }\xi _{ik,l^{\prime \prime }}\phi _{k,l^{\prime \prime }}\left( T_{ij}\right) X_{il^{\prime \prime }}\right\} _{l=1}^{d}\!\!, \end{aligned}$$

(38)

$$\begin{aligned} \hat{\varvec{\varepsilon }}(t)&= c_{J(t),n}^{-1/2}\mathbf {H} ^{-1}\left\{ \frac{1}{N_{{\mathrm{T}}}}\sum _{i=1}^{n}\sum \limits _{j=1}^{N_{i}}B_{J(t)}(T_{ij})X_{il}\sigma \left( T_{ij}\right) \varepsilon _{ij}\right\} _{l=1}^{d}\!\!. \end{aligned}$$

(39)

Proof of Proposition 4

Given $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$, let $\sigma _{\xi _{l},n}^{2}(t)$ and $\sigma _{\varepsilon _{l},n}^{2}(t)$ be the conditional variances of $\hat{\xi }_{l}(t)$ and $\hat{\varepsilon }_{l}(t)$ defined in ( 38) and (39), respectively. Define

$$\begin{aligned} \eta _{l}(t)=\left\{ \sigma _{\xi _{l},n}^{2}(t)+\sigma _{\varepsilon _{l},n}^{2}(t)\right\} ^{-1/2}\left\{ \hat{\xi }_{l}(t)+\hat{\varepsilon } _{l}(t)\right\} . \end{aligned}$$

(40)

By Lemma 7, $\eta _{l}(t)$ is a Gaussian process consisting of $\left( N_{{\mathrm{s}}}+1\right)$ standard normal variables $\left\{ \eta _{J,l}\right\} _{J=0}^{N_{{\mathrm{s}}}}$ such that $\eta _{l}(t)=\eta _{J(t),l}$ for $t\in \left[ 0,1\right]$. Thus, for any $\tau \in \mathbb {R}$,

$$\begin{aligned}&P\left( \sup _{t\in \left[ 0,1\right] }\left| \eta _{l}(t)\right| \le \tau /a_{N_{{\mathrm{s}}}\!+\!1}\!+\!b_{N_{{\mathrm{s}}}\!+\!1}\right) \\&\quad = P\left( |\max \{\eta _{0,l},\dots ,\eta _{N_{{\mathrm{s}}},l}\}|\le \tau /a_{N_{{\mathrm{s}}}\!+\!1}\!+\!b_{N_{ {\mathrm{s}}}+1}\right) \!. \end{aligned}$$

By Theorem 1.5.3 in Leadbetter et al. (1983), if $\xi _{0},\dots ,\xi _{N_{{\mathrm{s}}}}$ are i.i.d. standard normal r.v.’s, then for $\tau \in \mathbb {R}$

$$\begin{aligned} P\left( |\max \{\xi _{0},\dots ,\xi _{N_{{\mathrm{s}}}}\}|\le \tau /a_{N_{{\mathrm{s}} }}+b_{N_{{\mathrm{s}}}})\rightarrow \exp (-2\mathrm{e}^{-\tau }\right) . \end{aligned}$$

Next by Lemma 11.1.2 in Leadbetter et al. (1983),

$$\begin{aligned}&P\left( |\max \{\eta _{0,l},\dots ,\eta _{N_{{\mathrm{s}}},l}\}|\le \tau /a_{N_{ {\mathrm{s}}}+1}+b_{N_{{\mathrm{s}}}+1}\right) \\&\qquad \quad -P\left( |\max \{\xi _{0},\dots ,\xi _{N_{ {\mathrm{s}}}}\}|\le \tau /a_{N_{{\mathrm{s}}}+1}+b_{N_{{\mathrm{s}}}+1}\right) \\&\quad \le \frac{4}{2\pi }\sum _{0\le J<J^\prime \le N_{{\mathrm{s}}}}\!\!| {{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}|(1-| {{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}|^{2})^{-1/2}\exp \left\{ \frac{-(\tau /a_{N_{{\mathrm{s}}}+1}+b_{N_{{\mathrm{s}}}+1})^2}{1+{{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}}\right\} \!. \end{aligned}$$

According to Lemma 7, there exists a constant $C>0$ such that $\sup _{0\le J\ne J^{\prime }\le N_{{\mathrm{s}}}} \left| {{\mathsf {E}}}\eta _{J,l}\eta _{J^{\prime },l}\right| \le Ch_{{\mathrm{s}}}$ for large $n$. Thus, as $n\rightarrow \infty$,

$$\begin{aligned}&P\left( |\max \{\eta _{0,l},\dots ,\eta _{N_{{\mathrm{s}}},l}\}|\le \tau /a_{N_{{\mathrm{ s}}}+1}+b_{N_{{\mathrm{s}}}+1}\right) \\&\quad -P\left( |\max \{\xi _{0},\dots ,\xi _{N_{{\mathrm{s}} }}\}|\le \tau /a_{N_{{\mathrm{s}}}+1}+b_{N_{{\mathrm{s}}}+1}\right) \rightarrow 0. \end{aligned}$$

Therefore, for any $\tau \in \mathbb {R}$,

$$\begin{aligned} \lim \limits _{n\rightarrow \infty }P\left( \sup _{t\in \left[ 0,1\right] }\left| \eta _{l}(t)\right| \le \tau /a_{N_{{\mathrm{s}}}+1}+b_{N_{ {\mathrm{s}}}+1}\right) =\exp \left( -2\mathrm{e}^{-\tau }\right) . \end{aligned}$$

(41)

By Lemma 8, we have

$$\begin{aligned}&a_{N_{{\mathrm{s}}}+1}\left( \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \hat{\xi _{l}}(t)+\hat{\varepsilon } _{l}(t)\right| -\sup _{t\in \left[ 0,1\right] }\left| \eta _{l}(t)\right| \right) \\&\quad =\mathcal {O}_{p}\left\{ \log \left( N_{{\mathrm{s}}}+1\right) \left( nh_{{\mathrm{s}}}\right) ^{-1/2}(\log (n))^{1/2}\right\} ={ \scriptstyle {\mathcal {O}}}_{p}\left( 1\right) . \end{aligned}$$

On the other hand, Lemma 4 ensures that

$$\begin{aligned}&a_{N_{{\mathrm{s}}}+1}\left( \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \tilde{\xi }_{l}(t)+\tilde{\varepsilon } _{l}(t)\right| -\sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \hat{\xi _{l}}(t)+\hat{\varepsilon } _{l}(t)\right| \right) \\&\quad =\mathcal {O}_{p}\left\{ (\log \left( N_{{\mathrm{s}}}+1\right) nh_{{\mathrm{s}}})^{1/2}n^{-1}h_{{\mathrm{s}}}^{-3/2}\log (n)\right\} \\&\quad =\mathcal {O}_{p}\left\{ n^{-1/2}h_{{\mathrm{s}}}^{-1}(\log \left( N_{{\mathrm{s}}}+1\right) )^{1/2}\log (n)\right\} ={\scriptstyle { \mathcal {O}}}_{p}\left( 1\right) . \end{aligned}$$

Then the proof follows from (41) and Slutsky’s Theorem. $\square$

1.4 A.4 Proof of Theorem 1

For any vector $\mathbf {a}=\left( a_{1},\ldots ,a_{d}\right) ^{\scriptstyle { \mathsf {T}}}\in \mathbb {R}^{d}$, ${{\mathsf {E}}}\left[ \sum _{l=1}^{d}a_{l}\left\{ \hat{\xi }_{l}\left( t\right) +\hat{\varepsilon } _{l}\left( t\right) \right\} \right] =0.$ Using the conditional independence of $\hat{\xi }_{l}(t)$, $\hat{\varepsilon }_{l}(t)$ on $\left( T_{ij},N_{i},X_{il}\right) _{i=1,j=1,l=1}^{n,N_{i},d}$, we have

$$\begin{aligned}&\mathop {\text{ Var }}\left[ \left. \sum _{l=1}^{d}a_{l}\left\{ \hat{\xi } _{l}\left( t\right) +\hat{\varepsilon }_{l}\left( t\right) \right\} \right| \left( T_{ij},N_{i},X_{il}\right) _{j=1,i=1,l=1}^{N_{i},n,d} \right] \\&\quad =\sum _{l=1}^{d}\sum _{l^{\prime }=1}^{d}a_{l}a_{l^{\prime }} {{\mathsf {E}}}\left\{ \hat{\xi }_{l}\left( t\right) \hat{\xi } _{l^{\prime }}\left( t\right) +\hat{\varepsilon }_{l}\left( t\right) \hat{ \varepsilon }_{l^{\prime }}\left( t\right) \left| \left( T_{ij},N_{i},X_{il}\right) _{j=1,i=1,l=1}^{N_{i},n,d}\right. \right\} \\&\quad =\mathbf {a}^{\scriptstyle {\mathsf {T}}}\left\{ \mathbf {\Sigma }_{\xi ,n}(t)+ \mathbf {\Sigma }_{\varepsilon ,n}(t)\right\} \mathbf {a} ~. \end{aligned}$$

Meanwhile, Assumption (A2) entails that for any $t\in \left[ 0,1\right]$, given $\left( T_{ij},N_{i},X_{il}\right) _{j=1,i=1,l=1}^{N_{i},n,d}$, the conditional distribution of $\left[ \mathbf {a}^{\scriptstyle {\mathsf {T}} }\left\{ \mathbf {\Sigma }_{\xi ,n}(t)\!+\!\mathbf {\Sigma }_{\varepsilon ,n}(t)\right\} \mathbf {a}\right] ^{-1/2}\sum _{l=1}^{d}a_{l}\left\{ \hat{\xi } _{l}\left( t\right) +\hat{\varepsilon }_{l}\left( t\right) \right\}$ is a standard normal distribution. So we have

$$\begin{aligned} \left[ \mathbf {a}^{\scriptstyle {\mathsf {T}}}\left\{ \mathbf {\Sigma }_{\xi ,n}(t)+\mathbf {\Sigma }_{\varepsilon ,n}(t)\right\} \mathbf {a}\right] ^{-1/2}\sum _{l=1}^{d}a_{l}\left\{ \hat{\xi }_{l}\left( t\right) +\hat{ \varepsilon }_{l}\left( t\right) \right\} \sim N\left( 0,1\right) . \end{aligned}$$

Using (33), we have as $n\rightarrow \infty$

$$\begin{aligned} \left[ \mathbf {a}^{\scriptstyle {\mathsf {T}}}\mathbf {\Sigma }_{n}(t)\mathbf {a} \right] ^{-1/2}\sum _{l=1}^{d}a_{l}\left\{ \hat{\xi }_{l}\left( t\right) +\hat{ \varepsilon }_{l}\left( t\right) \right\} \overset{\mathcal {L}}{ \longrightarrow }N\left( 0,1\right) . \end{aligned}$$

Therefore, $\left[ \mathbf {a}^{\scriptstyle {\mathsf {T}}}\mathbf {\Sigma } _{n}(t)\mathbf {a}\right] ^{-1/2}\sum _{l=1}^{d}a_{l}\left\{ \hat{m}_{l}\left( t\right) -m_{l}\left( t\right) \right\} \overset{\mathcal {L}}{\longrightarrow }N\left( 0,1\right)$ follows from (24), Proposition 3, Lemma 4 and Slutsky’s Theorem. Applying Cramér–Wold’s device, we obtain $\mathbf {\Sigma } _{n}^{-1/2}\left( t\right) \left\{ \hat{m}_{l}\left( t\right) -m_{l}\left( t\right) \right\} _{l=1}^{d}\overset{\mathcal {L}}{\longrightarrow }N\left( \mathbf {0},\mathbf {I}_{d\times d}\right)$, and consequently, $\sigma _{n,ll}^{-1}(t)\left\{ \hat{m}_{l}(t)-m_{l}(t)\right\} \overset{\mathcal {L}}{ \longrightarrow } N\left( 0,1\right)$ for any $t\in \left[ 0,1\right]$ and $l=1,\ldots ,d.$ $\square$

1.5 A.5 Proof of Theorem 2

By Proposition 3, $\left\| \tilde{m} _{l}-m_{l}\right\| _{\infty }=\mathcal {O}_{p}\left( h_{{\mathrm{s}}}\right)$ , $l=1,\ldots , d$, so

$$\begin{aligned} a_{N_{{\mathrm{s}}}\!+\!1}\left\{ \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \tilde{m}_{l}(t)-m_{l}(t)\right| \right\} \!=\! \mathcal {O}_{p}\left\{ \left( nh_{{\mathrm{s}}}\right) ^{1/2}(\log \left( N_{{\mathrm{s}}}\!+\!1\right) )^{1/2}h_{{\mathrm{s}}}\right\} \!=\!{\scriptstyle { \mathcal {O}}}_{p}\left( 1\right) . \end{aligned}$$

According to (24), it is easy to show that

$$\begin{aligned} a_{N_{{\mathrm{s}}}+1}\left\{ \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \hat{m}_{l}(t)-m_{l}(t)\right| -\sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \tilde{\xi }_{l}(t)+ \tilde{\varepsilon }_{l}(t)\right| \right\} ={\scriptstyle {\mathcal {O}}} _{p}\left( 1\right) . \end{aligned}$$

Meanwhile, Proposition 4 entails that, for any $\tau \in \mathbb {R}$,

$$\begin{aligned} \lim _{n\rightarrow \infty }P\left\{ a_{N_{{\mathrm{s}}}+1}\left( \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \tilde{\xi }_{l}(t)+ \tilde{\varepsilon }_{l}(t)\right| -b_{N_{{\mathrm{s}}}+1}\right) \le \tau \right\} =\exp \left( -2\mathrm{e}^{-\tau }\right) . \end{aligned}$$

Thus Slutsky’s Theorem implies that

$$\begin{aligned} \lim _{n\rightarrow \infty }P\left\{ a_{N_{{\mathrm{s}}}+1}\left( \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \hat{m} _{l}(t)-m_{l}(t)\right| -b_{N_{{\mathrm{s}}}+1}\right) \le \tau \right\} =\exp \left( -2\mathrm{e}^{-\tau }\right) . \end{aligned}$$

Let $\tau =-\log \left\{ -\frac{1}{2}\log \left( 1-\alpha \right) \right\}$, the definition of $Q_{N_{{\mathrm{s}}}+1}\left( \alpha \right)$ in (9) entails

$$\begin{aligned}&\lim _{n\rightarrow \infty }P\left\{ m_{l}(t)\in \hat{m}_{l}(t)\pm \sigma _{n,ll}(t)Q_{N_{{\mathrm{s}}}+1}\left( \alpha \right) ,\forall t\in \left[ 0,1 \right] \right\} \\&\quad =\lim _{n\rightarrow \infty }P\left\{ \sup _{t\in \left[ 0,1\right] }\sigma _{n,ll}^{-1}(t)\left| \hat{m}_{l}(t)-m_{l}(t)\right| \le Q_{N_{ {\mathrm{s}}}+1}\left( \alpha \right) \right\} =1-\alpha . \end{aligned}$$

Theorem 2 is proved. $\square$