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\begin{document}





% Title of paper
\title{Mark-Specific Proportional Hazards Model with Multivariate Continuous Marks
and Its Application to HIV Vaccine Efficacy Trials}
  

\author{Yanqing Sun\\
Department of Mathematics and Statistics\\
University of North Carolina at Charlotte, Charlotte, NC 28223 \\
Mei Li$^\ast$\\
School of Public Health\\
Zhejiang University, Hangzhou, China\\
Peter B.\ Gilbert\\
Department of Biostatistics, University of Washington\\
and Fred Hutchinson Cancer Research Center,
Seattle, WA 98109}



% Running headers of paper:
\markboth%
% First field is the short list of authors
{Y. Sun, M. Li and P. B. Gilbert}
% Second field is the short title of the paper
{Proportional hazards model with multivariate continuous marks}

\maketitle

% Add a footnote for the corresponding author if one has been
% identified in the author list
\footnotetext{To whom correspondence should be addressed.}

%\date{}

\vspace{.55cm}
 \centerline{\textsc Supplementary Material}
\vspace{.55cm}

%\section*{Supplementary Material}

%CONTENT OF A BRIEF NOTE.........\\
%CONTENT OF THE BRIEF NOTE.\\
%CONTENT OF THE BRIEF NOTE.\\

This Supplementary Material presents the proofs for Theorem 2.1, Theorem 2.2 and Theorem 3.1 of the paper.
Please refer to the main paper for notations not defined here.


\par

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\bigskip


\begin{lemmaS}
\label{converge}Under conditions (A.1)-(A.3),
$S_k^{(j)}(t,v,\bar\beta)$ converges to $s_k^{(j)}(t,v,\bar \beta)$
in probability  uniformly in $(t,v,\bar\beta)\in [0,\tau]\times [0,1]^2\times\
{\cal B}$ as $n \to \infty$, for $j=0,1,2$ and $1 \leq k \leq
K$.
\end{lemmaS}

\noindent  \textsc{Proof of Lemma \ref{converge}.}

 Let $S_{ki}^{(j)}(t,v,\bar \beta)=Y_{ki}(t)\exp\{\bar{\beta}^T\tilde
Z_{ki}(t,v)\}\tilde Z_{ki}(t,v)^{\otimes{j}}$. Then
$S_k^{(j)}(t,v,\bar \beta)=n_k^{-1}\sum_{i=1}^{n_k}S_{ki}^{(j)}(t,v,\bar \beta)$.
We prove the lemma for the case when $j=0$. The proofs for $j=1$ and $2$ follow
similarly. Let $\omega_{ki}=(X_{ki},Z_{ki})$. Then $\omega_{ki}$,
$i=1, \ldots, n_k$, is a random sample from a probability
distribution $ P_k $ on a measurable space $({\cal X}_k, {\cal
A}_k)$, where ${\cal X}_k=[0,\tau] \times R^p$
and ${\cal A}_k$ is its Borel $\sigma$-field. Let ${\cal F}$ be the
class of all coordinate projections $f_{t,v,\bar
\beta}(\omega_{ki}):\; {\cal X}_k\longrightarrow R$, where
$f_{t,v,\bar \beta}(\omega_{ki})=S_{ki}^{(0)}(t,v,\bar \beta)$, for $(t,v,\bar \beta)
\in [0,\tau] \times [0,1]^2 \times \cal B$. Then $S_k^{(0)}(t,v,\bar
\beta)=n_k^{-1}\sum_{i=1}^{n_k }f_{t,v,\bar \beta}(\omega_{ki})$.
Let $\|f_{t,v,\bar \beta}\|_{P_k,r}=( P_k |f_{t,v,\bar
\beta}|^r)^{1/r}=(E_k|S_{ki}^{(0)}(t,v,\bar \beta)|^r)^{1/r}$ be $L_r( P_k
)$-norm of $f_{t,v,\bar\beta}$. Next, we show that $\cal F$ is
Glivenko-Cantelli (van der Vaart, 1998). Since $Z_{ki}(\cdot)$ is of bounded variation,
for simplicity we assume that $Z_{ki}(\cdot)$ is an nonnegative
monotone increasing process. In general, $Z_{ki}(\cdot)$ can be
expressed as the difference of two nonnegative monotone increasing
processes plus a constant. In this case, the class of functions of
interest, $\cal F$, is the product of several functional classes.
It is Glivenko-Cantelli if each of them is Glivenko-Cantelli.

 Let $\{t_h\}$, $\{v_j\}$ and $\{\bar \beta_m\}$ be the grid
points of finite partitions of the intervals $[0,\tau]$, $[0,1]^2$,
and ${\cal B}$, respectively. Let $\{t_{h^{'}},t_h\}$,
$\{v_{j^{'}},v_j\}$ and $\{\bar \beta_{m^{'}},\bar\beta_m\}$ be the
grid points on the opposite ends of a hyper-cubic of the partitions
such that $0\leq t_h-t_{h^{'}}\leq \epsilon$, $0\leq
v_j-v_{j^{'}}\leq \epsilon$ and $0\leq \bar \beta_m-\bar
\beta_{m^{'}}\leq \epsilon$ for $\epsilon > 0$. Define the
bracketing functions $l_{h^{'}j^{'}m^{'}}=S_{ki}^{(0)}(t_{h^{'}},
v_{j^{'}},\bar \beta_{m^{'}})$ and $u_{hjm}=S_{ki}^{(0)}(t_{h},
v_{j},\bar \beta_{m})$. Then for any $f_{t,v,\bar \beta} \in {\cal
F}$, there is a bracket $[l_{h^{'}j^{'}m^{'}},u_{hjm}]$ such that
$f_{t,v,\bar \beta} \in [l_{h^{'}j^{'}m^{'}},u_{hjm}]$. Further,

\begin{eqnarray*}
\label{cc1} \|u_{hjm}-l_{h^{'}j^{'}m^{'}}\|_{P_k,2} & & \le
\|S_{ki}^{(0)}(t_{h},v_{j},\bar\beta_{m})-S_{ki}^{(0)}
(t_{h^{'}},v_{j^{'}},\bar\beta_{m^{'}})\|_{P_k,2}\\
& &=\|Y_{ki}(t_h)exp\{\bar{\beta}_{m}^T\tilde Z_{ki}(t_h,v_j)\}
 -Y_{ki}(t_{h^{'}})exp\{\bar{\beta}_{m^{'}}^T\tilde
Z_{ki}(t_{h^{'}},v_{j^{'}})\}\|_{P_k,2}\\
 & & \le
[C_1\|t_h-t_{h^{'}}\|+C_2\|v_{j}-v_{j^{'}}\|+C_3\|\bar\beta_m-\bar\beta_{m^{'}}\|]^{1/2}\\
& & \leq C\epsilon^{1/2},
\end{eqnarray*}
where $C_1$, $C_2$, $C_3$ and $C$ are some positive constants.
Hence, the bracketing number $N_{[\, ]}(\epsilon^{1/2},{\cal F},L_2(
P_k ))$ is of the polynomial order $(1/\epsilon)^{4p+3}$. Thus
$N_{[\, ]}(\epsilon,{\cal F},L_2( P_k ))$ is of the order
$(1/\epsilon)^{2(4p+3)}$. By the Glivenko--Cantelli Theorem (Theorem
19.4 of van der Vaart, 1998), $S_k^{(0)}(t,v,\bar\beta)$ converges in
probability uniformly to $s_k^{(0)}(t,v,\bar\beta)$ for
$(t,v,\bar\beta)\in [0,\tau]\times [0,1]^2\times {\cal B}$. \qquad\qed

\bigskip


\noindent \textsc{Proof of Theorem 2.1.}


Let
\begin{eqnarray*}
\label{ll}
\eta_n(\bar\beta)&=&n^{-1}(l(\bar\beta)-l(\bar\beta_0))\\
&=&n^{-1}\sum_{k=1}^K\sum_{i=1}^{n_k}\int_0^\tau\int_0^1\bigg[(\bar\beta-\bar\beta_0)^{T}\tilde
          Z_{ki}(s,u)-\log\bigg\{\frac{S_{k}^{(0)}(s,u,\bar\beta)}
          {S_{k}^{(0)}(s,u,\bar\beta_0)}\bigg\}\bigg]N_{ki}(ds,du).
\end{eqnarray*}
By the condition (A.2), Lemma \ref{converge} and the uniform convergence of
$n_k^{-1}\sum_{i=1}^{n_k} N_{ki}(t,v)\gop$ $ \int_0^t\int_0^v s_k^{(0)}(s,u,\bar\beta_0)\lambda_{0k}(s,u)\,dsdu$
(Gilbert, Mckeague and Sun, 2004),
we have $\eta_n(\bar\beta) \gop \eta(\bar\beta)$, where
$$\eta(\bar\beta) =\sum_{k=1}^K p_k E\bigg(\int_0^\tau\int_0^1\bigg[(\bar\beta-\bar\beta_0)^{T}\tilde
          Z_{ki}(s,u)-\log\bigg\{\frac{s_{k}^{(0)}(s,u,\bar\beta)}
          {s_{k}^{(0)}(s,u,\bar\beta_0)}\bigg\}\bigg]\bigg)s_k^{(0)}(s,u,\bar\beta_0)\lambda_{0k}(s,u)\,dsdu,$$
uniformly in $\bar\beta\in {\cal B}$. Further,
$-{\partial^2 \eta_n(\bar \beta)}/{\partial \bar\beta^2}=n^{-1}I(\bar\beta)$
converges in probability to $\Sigma(\bar\beta)$
uniformly in $\bar\beta \in \cal B$. The limiting
matrix function is a positive definite matrix for $\bar\beta\in {\cal B}$
under the conditions (A.2) and (A.3). Hence, $\eta_n(\bar\beta)$ converges in probability to $\eta(\bar\beta)$
which is a concave function with a unique maximum at
$\bar \beta_0$. Since $\hat{\bar\beta}$ is the maximizer of $\eta_n(\bar\beta)$, we
have $\hat{\bar \beta}$ converges in probability to $\bar \beta_0$
as $n\to \infty$, see van der Vaart (1998). \qquad\qed



\bigskip
\noindent \textsc{Proof of Theorem 2.2.}

 Note that
$U(\hat{\bar\beta})-U(\bar{\beta}_0)=-I(\bar{\beta}^*)(\hat{\bar\beta}-\bar{\beta}_0)$,
where $\bar{\beta}^*$ is on the line segment between
$\hat{\bar\beta}$ and $\bar{\beta}_0$. By the uniform convergence of
$n^{-1}I(\bar\beta)\gop \Sigma(\bar\beta)$ in probability in $\bar\beta
\in \cal B$ and the consistency of $\hat{\bar\beta}$ to
$\bar\beta_0$, we have
\begin{eqnarray}
\label{taylor} & &n^{1/2}(\hat{\bar
\beta}-\bar{\beta}_0)=(I(\bar{\beta}^*)/n)^{-1}
n^{-1/2}U(\bar{\beta}_0) \nonumber\\
& & = (\Sigma(\bar\beta_0))^{-1} n^{-1/2} U(\bar{\beta}_0)+o_p(1). \label{Taylor1}
\end{eqnarray}
It remains to show that $n^{-1/2}U(\bar \beta_0)\god
N(0,\Sigma(\bar \beta_0))$. Observe that
\begin{eqnarray}
\label{uu} n^{-1/2}U(\bar{\beta}_0)
&=&n^{-1/2}\sum_{k=1}^K\sum_{i=1}^{n_k}\int_0^\tau\int_0^1\bigg[\tilde{Z}_{ki}(t,u)-
    \frac{S_k^{(1)}(t,u,\bar{\beta}_0)}{S_k^{(0)}(t,u,\bar{\beta}_0)}\bigg]M_{ki}(dt,du).
    \label{Score-Martingale1}
\end{eqnarray}
By  Lemma \ref{converge} of this Appendix and Lemma 2 of Gilbert, Mckeague and Sun (2008),
\begin{eqnarray}
n^{-1/2}U(\bar{\beta}_0)
&=&n^{-1/2}\sum_{k=1}^K\sum_{i=1}^{n_k}\int_0^\tau\int_0^1\bigg[\tilde{Z}_{ki}
   (t,u)-\frac{s_k^{(1)}(t,u,\bar{\beta})}{s_k^{(0)}(t,u,\bar{\beta})}\bigg]M_{ki}(dt,du)+o_p(1),
     \label{Score-Martingale2}
\end{eqnarray}
which converges in distribution to $N(0,\Sigma(\bar{\beta}_0))$. This completes the proof.
\qquad\qed

\bigskip

\noindent \textsc{Proof of Theorem 2.3.}

Note that 
\begin{eqnarray}
& &\hat\Lambda_{0k}(t,v)-\Lambda_{0k}(t,v)\nonumber\\
& &=\int_0^t\int_0^v \bigg[
\frac{1}{n_k S_k^{(0)}(s,u,\hat{\bar\beta})}-\frac{1}{n_k S_k^{(0)}(s,u,\bar\beta_0)}\bigg] N_{k\cdot}(ds,du)+\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2})\nonumber\\
& &=\int_0^t\int_0^v \frac{[S_k^{(1)}(s,u,\bar\beta_0)]^T(\bar\beta_0-\hat{\bar\beta})
}{n_k S_k^{(0)}(s,u, \hat{\bar\beta})S_k^{(0)}(s,u,\bar\beta_0)}\,N_{k\cdot}(ds,du)
+\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2})\nonumber\\
& &=\int_0^t\int_0^v \frac{[S_k^{(1)}(s,
u,\bar\beta_0)]^T(\bar\beta_0-\hat{\bar\beta})
}{S_k^{(0)}(s, u,\hat{\bar\beta})}\lambda_{0k}(s,u)\,dsdu +\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k
S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2}), \label{decomp_lambda}
\end{eqnarray}
where $M_{k\cdot}(t,v)=\sum_{i=1}^{n_k}M_{ki}(t,v)$. 
By Slusky Theorem, Lemma 2 of Gilbert, Mckeague and Sun (2008),
Lemma A.2 of Sun et al.\ (2009), we have
\begin{eqnarray*}
& &\sqrt n \{\hat\Lambda_{0k}(t,v)-\Lambda_{0k}(t,v)\} \nonumber\\
& &=-\sqrt n (\hat{\bar\beta}-\bar\beta_0) \int_0^t\int_0^v \frac{[s_k^{(1)}(s,
u,\bar\beta_0)]
}{s_k^{(0)}(s, u,\bar\beta_0)}\lambda_{0k}(s,u)\,dsdu +\sqrt n \int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k
s_k^{(0)}(s,u,\bar\beta_0)}+o_p(1). \label{decomp_lambda2}
\end{eqnarray*}
Since $M_{ki}(t,v)$'s are mark-specific martingales, by (\ref{Taylor1}) and
(\ref{Score-Martingale1}), it is easy to check that 
$\sqrt n(\hat{\bar\beta}-\bar\beta_0)$ is asymptotically independent of
the processes $\sqrt n\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k
s_k^{(0)}(s,u,\bar\beta_0)}$, $k=1,\ldots,K$. 
Following the proof of Lemma A.2 of Sun et al.\ (2009), it can be shown that
the latter are asymptotically independent
mean-zero Guassian random fields, each of them has independent increments. 
\qquad\qed


\noindent \textsc{ Proof of Theorem 3.1.}

The cumulative martingale residuals are defined as
$$ \hat M_{ki}(t,v)=\int_0^t\int_0^v
N_{ki}(ds,du)-Y_{ki}(s) \exp(\hat{\bar \beta}^T \tilde Z_{ki}(s,u))
\hat\Lambda_{0k}(ds,du),
$$
where $\hat \Lambda_{0k}(t,v)=\int_0^t\int_0^v
\{n_k S_k^{(0)}(s,u,\hat{\bar \beta})\}^{-1}{N_{k\cdot}(ds,du)}
$ is the estimator of the doubly cumulative baseline function
$\Lambda_{0k}(t,v)=\int_0^t\int_0^v \lambda_{0k}(s,u)dsdu$.
It follows that
\begin{eqnarray}
\hat M_{ki}(t,v)&=&M_{ki}(t,v)-\int_0^t\int_0^v
Y_{ki}(s) \exp(\hat{\bar\beta}^T\tilde Z_{ki}(s,u)) [\hat\Lambda_{0k}(ds,du)-\Lambda_{0k}(ds,du)]\nonumber\\
& &  -\int_0^t\int_0^v Y_{ki}(s) [\exp(\hat{\bar\beta}^T\tilde
Z_{ki}(s,u)) - \exp(\bar{\beta}^T_0\tilde
Z_{ki}(s,u)]\Lambda_{0k}(ds,du)]. \label{decomp_res1}
\end{eqnarray}
\comment{
Note that 
\begin{eqnarray}
& &\hat\Lambda_{0k}(t,v)-\Lambda_{0k}(t,v)\nonumber\\
& &=\int_0^t\int_0^v \bigg[
\frac{1}{n_k S_k^{(0)}(s,u,\hat{\bar\beta})}-\frac{1}{n_k S_k^{(0)}(s,u,\bar\beta_0)}\bigg] N_{k\cdot}(ds,du)+\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2})\nonumber\\
& &=\int_0^t\int_0^v \frac{[S_k^{(1)}(s,u,\bar\beta_0)]^T(\bar\beta_0-\hat{\bar\beta})
}{n_k S_k^{(0)}(s,u, \hat{\bar\beta})S_k^{(0)}(s,u,\bar\beta_0)}\,N_{k\cdot}(ds,du)
+\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2})\nonumber\\
& &=\int_0^t\int_0^v \frac{[S_k^{(1)}(s,
u,\bar\beta_0)]^T(\bar\beta_0-\hat{\bar\beta})
}{S_k^{(0)}(s, u,\hat{\bar\beta})}\lambda_{0k}(s,u)\,dsdu +\int_0^t\int_0^v \frac{M_{k\cdot}(ds,du)}{n_k
S_k^{(0)}(s,u,\bar\beta_0)}+o_p(n_k^{-1/2}), \label{decomp_lambda}
\end{eqnarray}
where $M_{k\cdot}(t,v)=\sum_{i=1}^{n_k}M_{ki}(t,v)$. 
}
By (\ref{decomp_lambda}), we have
\begin{eqnarray}
W_k(t,v,z)&=& n^{-1/2} \sum_{i=1}^{n_k} g_k(Z_{ki},z) M_{ki}(t,v)\nonumber\\
& & +n_k n^{-1/2} \int_0^t\int_0^v \frac{S_{kg}^{(0)}(s,u,z,
\hat{\bar\beta}) [S_k^{(1)}(s,u, \bar\beta_0)]^T
}{S_k^{(0)}(s,u,\bar\beta_0)}  \lambda_{0k}(s,u) \,dsdu \,(\hat{\bar\beta}-\bar\beta_0) \nonumber\\
& & -n^{-1/2}\int_0^t\int_0^v \frac{S_{kg}^{(0)}(s,u,z,\hat{\bar\beta})}
{ S_k^{(0)}(s,u,\bar\beta_0)}\,M_{k\cdot}(ds,du)\nonumber\\
& &-n_k n^{-1/2}\int_0^t\int_0^v
[S_{kg}^{(1)}(s, u,z,\bar\beta_0)]^T
\lambda_{0k}(s,u)\,dsdu \,(\hat{\bar\beta}-\bar\beta_0) +o_p(1).
\label{decomp1_res}
\end{eqnarray}
Replacing $\hat{\bar\beta}$ in $S_{kg}^{(0)}(s,u,z,
\hat{\bar\beta})$ by $\bar\beta_0$ results in a process that is
equivalent to  $W_k(t,v,z)$ in probability by Slusky Theorem for
the second term and by Lemma 2 of Gilbert, Mckeague and Sun (2008) for the third term of (\ref{decomp1_res}).

From the proof of Theorem 2, 
we have
$n^{1/2}(\hat{\bar\beta}-\bar\beta_0)=(\Sigma(\bar{\beta}_0))^{-1}
n^{-1/2} U(\bar\beta_0)+o_p(1)$. The second term of (\ref{decomp1_res}) equals
$$ (n_k/n)\bigg\{\int_0^t\int_0^v \frac{[S_k^{(1)}
 (s,u,\bar{\beta}_0)\otimes S_{kg}^{(0)}(s,u,z,\bar \beta)]^T}{S_k^{(0)}(s,u,\bar{\beta}_0)}\lambda_{0k}(s,u)\,dsdu\bigg\}(\Sigma(\bar{\beta}_0))^{-1}
 n^{-1/2} U(\bar\beta_0)+o_p(1)$$
Similarly, the fourth term of (\ref{decomp1_res}) is equal to
\begin{eqnarray*}
& & (n_k/n)\bigg\{\int_0^t\int_0^v
{[S_{kg}^{(1)}(s,u,z,\bar{\beta}_0)]^T\lambda_{0k}(s,u)}\,dsdu\bigg\}
(\Sigma(\bar{\beta}_0))^{-1} n^{-1/2} U(\bar\beta_0) +o_p(1).
\end{eqnarray*}
Bringing the above expressions into (\ref{decomp1_res}), we have
\begin{eqnarray}
& & W_k(t,v,z)\nonumber \\
& & =n^{-1/2} \sum_{i=1}^{n_k} \int_0^t\int_0^v \bigg
[g_k(Z_{ki},z)-\frac{S_{kg}^{(0)}(s,u,z,\bar{\beta}_0) }
{ S_k^{(0)}(s,u,z,\bar{\beta}_0)}\bigg ]\,M_{ki}(ds,du)\nonumber\\
& &\quad + {n_k}/{n}\int_0^t \int_0^v
 \bigg(\frac{S_k^{(1)}(s,u,\bar{\beta}_0) \otimes
S_{kg}^{(0)}(s,u,z,\bar{\beta}_0) }{S_k^{(0)}(s,u,\bar{\beta}_0)} -
S_{kg}^{(1)}(s,u,z,\bar{\beta}_0)\bigg)^T\lambda_{0k}(s,u)\,dsdu\nonumber\\
& &\quad (\Sigma(\bar{\beta}_0))^{-1} n^{-1/2} U(\bar\beta_0) +o_p(1)\nonumber\\
& &= n^{-1/2}\sum_{i=1}^{n_k} \int_0^t\int_0^v \bigg
[g_k(Z_{ki},z)-\frac{S_{kg}^{(0)}(s,u,z, \bar{\beta}_0) }
{ S_k^{(0)}(s,u,\bar{\beta}_0)}\bigg ]\,M_{ki}(ds,du)\nonumber \\
& & \quad +(R_k(t,v,z))^T (\Sigma(\bar{\beta}_0))^{-1} n^{-1/2}\sum_{l=1}^K\sum_{i=1}^{n_l}
\bigg\{\int_0^\tau\int_0^1 \bigg [\tilde Z_{li}(s,u)-
\frac{S_l^{(1)}(s,u,\bar{\beta}_0)}{S_l^{(0)}(s,u, \bar{\beta}_0)}
\bigg]M_{li}(ds,du)\bigg\}^T \nonumber \\
& & \quad +o_p(1)\nonumber\\
& &= n^{-1/2}\sum_{l=1}^K\sum_{i=1}^{n_l} \int_0^\tau\int_0^v
I(l=k)I(s\le t) \bigg [g_l(Z_{li},z)-\frac{S_{lg}^{(0)}(s,u,z,
\bar{\beta}_0) }
{ S_l^{(0)}(s,u,\bar{\beta}_0)}\bigg ]\,M_{li}(ds,du) \nonumber\\
& & \quad +(R_k(t,v,z))^T (\Sigma(\bar{\beta}_0))^{-1}
n^{-1/2}\sum_{l=1}^K\sum_{i=1}^{n_l} \bigg\{\int_0^\tau\int_0^1
 \bigg [\tilde Z_{li}(s,u)-
\frac{S_l^{(1)}(s,u,\bar{\beta}_0)}{S_l^{(0)}(s,u, \bar{\beta}_0)}
\bigg]M_{li}(ds,du)\bigg\}^T  \nonumber\\
& & \quad +o_p(1).
 \label{decomp3_res}
\end{eqnarray}
By the uniform convergence of $S_k^{(0)}(s,u,\bar{\beta}_0)$,
$S_k^{(1)}(s,u,\bar{\beta}_0)$, $S_{kg}^{(0)}(s,u,z,\bar{\beta}_0)$ and
$S_{kg}^{(1)}(s,u,z,\bar{\beta}_0)$ to  $s_k^{(0)}(s,u,\bar{\beta}_0)$,
$s_k^{(1)}(s,u,\bar{\beta}_0)$, $s_{kg}^{(0)}(s,u,z,\bar{\beta}_0)$ and
$s_{kg}^{(1)}(s,u,z,\bar{\beta}_0)$ in $(s,u)\in [0,\tau]\times[0,1]^2$
in probability, respectively, and by the weak convergence of
$n_k^{-1/2}\sum_{i=1}^{n_k}M_{ki}(t,v)$
for $k=1,\ldots K,$ the terms $S_k^{(0)}(s,u,\bar{\beta}_0)$,
$S_k^{(1)}(s,u,\bar{\beta}_0)$, $S_{kg}^{(0)}(s,u,z,\bar{\beta}_0)$ and
$S_{kg}^{(1)}(s,u,z,\bar{\beta}_0)$ can be replaced by their expected values respectively.
This completes the proof. \qquad\qed

\end{document}
