$\displaystyle Q_{3}=$ $\displaystyle
U(\beta,\pi^{*},S^{*},\hat{S}_{c})-U(\beta,\pi^{*},S^{*},S^{*}_{c}).$ (124)

We now show that $Q_{1},Q_{2}$, and $Q_{3}$ are $o_{p}(1)$.

Consider $Q_{1}$. We write

$\displaystyle Q_{1}=Q_{11}-Q_{12}-Q_{13}$ (125) $\displaystyle Q_{11}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(1)}(t;\pi^{*},\hat{S},\hat{S}_{c})$
(126) $\displaystyle Q_{12}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\left\\{\bar{A}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right\\}d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})$
(127) $\displaystyle Q_{13}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\left\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\pi^{*},\hat{S},\hat{S}_{c})\right\\}.$
(128)

First, we note that $d\mathcal{N}_{i}^{(1)}(t;\pi,S,S_{c})$ is a sum of
several terms, each term is a product of a term that is bounded a.s. and an
increment of a monotone function. Specifically, we have

$\displaystyle Q_{11}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left[\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}\right]\int_{0}^{\tau}\frac{A_{i}}{\hat{S}_{c}(t;A_{i},Z_{i})}dN_{i}(t)$
(129)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left[\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}\right]\int_{0}^{\tau}A_{i}d\hat{S}(t;A_{i},Z_{i})$
(130)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}A_{i}\left\\{\frac{1}{\hat{\pi}(Z_{i})}-\frac{1}{\pi^{*}(Z_{i})}\right\\}\int_{0}^{\tau}J_{i}(t;1,\hat{S},\hat{S}_{c})d\hat{S}(t;1,Z_{i}).$
(131)

This allows us to make use of the following property: for any function $f(t)$,
and any monotone function $G(t)$ defined on $[a,b]$, we have

$\displaystyle\left|\int_{a}^{b}f(t)dG(t)\right|\leq\sup_{t\in[a,b]}|f(t)|\cdot|G(b)-G(a)|.$
(132)

Since $N(t)$ and $\hat{S}(t;a,z)$ are monotone in $t$, we apply (132) to each
of the 3 terms in $Q_{11}$ above and have

$\displaystyle|Q_{11}|$ (133) $\displaystyle\leq$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left|\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}\right|\cdot\sup_{t\in[0,\tau]}\left|\frac{A_{i}}{\hat{S}_{c}(t;A_{i},Z_{i})}\right|\cdot\left|N_{i}(\tau)-N_{i}(0)\right|$
(134)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\left|\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}\right|\cdot|A_{i}|\cdot\left|\hat{S}(\tau;A_{i},Z_{i})-\hat{S}(0;A_{i},Z_{i})\right|$
(135)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}|A_{i}|\left|\frac{1}{\hat{\pi}(Z_{i})}-\frac{1}{\pi^{*}(Z_{i})}\right|\cdot\sup_{t\in[0,\tau]}\left|J_{i}(t;1,\hat{S},\hat{S}_{c})\right|\cdot\left|\hat{S}(\tau;1,Z_{i})-\hat{S}(0;1,Z_{i})\right|.$
(136) $\displaystyle\leq$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\frac{|\hat{\pi}(Z_{i})-\pi^{*}(Z_{i})|}{\left|\\{\hat{\pi}(Z_{i})\pi^{*}(Z_{i})\\}^{A_{i}}[\\{1-\hat{\pi}(Z_{i})\\}\\{1-\pi^{*}(Z_{i})\\}]^{1-A_{i}}\right|}\cdot\sup_{t\in[0,\tau]}\left|\frac{A_{i}}{\hat{S}_{c}(t;A_{i},Z_{i})}\right|\cdot\left|N_{i}(\tau)-N_{i}(0)\right|$
(137)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\frac{|\hat{\pi}(Z_{i})-\pi^{*}(Z_{i})|}{\left|\\{\hat{\pi}(Z_{i})\pi^{*}(Z_{i})\\}^{A_{i}}[\\{1-\hat{\pi}(Z_{i})\\}\\{1-\pi^{*}(Z_{i})\\}]^{1-A_{i}}\right|}\cdot|A_{i}|\cdot\left|\hat{S}(\tau;A_{i},Z_{i})-\hat{S}(0;A_{i},Z_{i})\right|$
(138)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}|A_{i}|\frac{|\hat{\pi}(Z_{i})-\pi^{*}(Z_{i})|}{|\hat{\pi}(Z_{i})\pi^{*}(Z_{i})|}\cdot\sup_{t\in[0,\tau]}\left|J_{i}(t;1,\hat{S},\hat{S}_{c})\right|\cdot\left|\hat{S}(\tau;1,Z_{i})-\hat{S}(0;1,Z_{i})\right|.$
(139)

Since $\hat{S}(t;a,z)$ and $\hat{S}_{c}(t;a,z)$ are bounded away from zero
a.s., we can again apply (132) to

$\displaystyle
J_{i}(t;1,\hat{S},\hat{S}_{c})=\int_{0}^{t}\frac{dN_{ci}(u)+Y_{i}(u)d\log\\{\hat{S}_{c}(u;1,Z_{i})\\}}{\hat{S}(u;1,Z_{i})\hat{S}_{c}(u;1,Z_{i})},$
(140)

and have

$\displaystyle\sup_{t\in[0,\tau]}\left|J_{i}(t;1,\hat{S},\hat{S}_{c})\right|\leq$
$\displaystyle\sup_{t\in[0,\tau]}\Bigg{\\{}\sup_{u\in[0,t]}\bigg{|}\frac{1}{\hat{S}(u;1,Z_{i})\hat{S}_{c}(u;1,Z_{i})}\bigg{|}\cdot\left|N_{ci}(t)-N_{ci}(0)\right|\Bigg{\\}}$
(141)
$\displaystyle+\sup_{t\in[0,\tau]}\Bigg{\\{}\sup_{u\in[0,t]}\bigg{|}\frac{Y_{i}(u)}{\hat{S}(u;1,Z_{i})\hat{S}_{c}(u;1,Z_{i})}\bigg{|}\cdot\left|\log\\{\hat{S}_{c}(t;1,Z_{i})\\}-\log\\{\hat{S}_{c}(0;1,Z_{i})\\}\right|\Bigg{\\}}$
(142) $\displaystyle\lesssim$ $\displaystyle 1.$ (143)

In addition, since $\hat{\pi}(z)$ and $1-\hat{\pi}(z)$ are bounded away from
zero a.s., we have

$\displaystyle|Q_{11}|\lesssim\frac{1}{n}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{*}(Z_{i})|.$
(144)

As a reminder, $E^{\dagger}$ denotes expectations taken with respect to a
sample $O^{\dagger}$ of $n$ observations, and $E$ denotes expectations taken
with respect to an independent data $O$ conditional on $O^{\dagger}$. $O$ is
used for constructing $U$, while $O^{\dagger}$ is used to estimate
$(\hat{\pi},\hat{S},\hat{S}_{c})$. Using this notation, we have

$\displaystyle E(|Q_{11}|)\lesssim$ $\displaystyle
E^{\dagger}\left[E\left\\{|\hat{\pi}(Z)-\pi^{*}(Z)|\right\\}\right]\leq\|\hat{\pi}-\pi^{*}\|_{\dagger}=o(1),$
(145)

where the last inequality follows from Jensen’s inequality, and the last
equality follows from Assumption 6. So we have $Q_{11}=o_{p}(1)$ from Markov’s
inequality.

Consider $Q_{12}$. We again break $d\mathcal{N}_{i}^{(0)}(t;\pi,S,S_{c})$ into
a sum of terms, each being a product of a term that is bounded a.s. and an
increment of a monotone function.

$\displaystyle Q_{12}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\left\\{\bar{A}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right\\}\cdot\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}\hat{S}_{c}(t;A_{i},Z_{i})}dN_{i}(t)$
(146)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\left\\{\bar{A}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right\\}\cdot\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}d\hat{S}(t;A_{i},Z_{i})$
(147)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\left\\{\bar{A}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right\\}$
(148)
$\displaystyle\quad\cdot\sum_{a=0,1}\left\\{1+\frac{A_{i}^{a}(1-A_{i})^{1-a}}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}J_{i}(t;a,\hat{S},\hat{S}_{c})\right\\}d\hat{S}(t;a,Z_{i}).$
(149)

Applying (132) and similar arguments as the above, also recall that
$\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})$ and
$\mathcal{S}^{(0)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})$ are bounded away from
zero a.s. and $\mathcal{S}^{(l)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})$ is
bounded a.s., we have

$\displaystyle|Q_{12}|\lesssim$
$\displaystyle\sup_{t\in[0,\tau]}\left|\bar{A}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right|$
(150) $\displaystyle=$
$\displaystyle\sup_{t\in[0,\tau]}\left|\frac{\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(1)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})-\mathcal{S}^{(1)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})}{\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})}\right|$
(151) $\displaystyle\lesssim$
$\displaystyle\sup_{t\in[0,\tau]}\left|\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(1)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})-\mathcal{S}^{(1)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right|$
(152) $\displaystyle\leq$
$\displaystyle\sup_{t\in[0,\tau]}\left|\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\\{\mathcal{S}^{(1)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})-\mathcal{S}^{(1)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\\}\right|$
(153)
$\displaystyle+\sup_{t\in[0,\tau]}\left|\mathcal{S}^{(1)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\\{\mathcal{S}^{(0)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})-\mathcal{S}^{(0)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})\\}\right|$
(154) $\displaystyle\lesssim$
$\displaystyle\sum_{l=0,1}\sup_{t\in[0,\tau]}\left|\mathcal{S}^{(l)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\mathcal{S}^{(l)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right|$
(155) $\displaystyle\leq$
$\displaystyle\sum_{l=0,1}\cdot\frac{1}{n}\sum_{i=1}^{n}\sup_{t\in[0,\tau]}\left|\Gamma_{i}^{(l)}(t;\beta,\hat{\pi},\hat{S},\hat{S}_{c})-\Gamma_{i}^{(l)}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})\right|$
(156) $\displaystyle\lesssim$
$\displaystyle\sum_{l=0,1}\cdot\frac{1}{n}\sum_{i=1}^{n}\left|\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}\right|$
(157)
$\displaystyle+\sum_{l=0,1}\cdot\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}a^{l}\left|\frac{1}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}-\frac{1}{\pi^{*}(Z_{i})^{a}\\{1-\pi^{*}(Z_{i})\\}^{1-a}}\right|$
(158) $\displaystyle\lesssim$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{*}(Z_{i})|,$
(159)

where (158) follows since $\hat{S}_{c}(t;A_{i},Z_{i})$ is bounded away from
zero a.s. and $J_{i}(t;a,\hat{S},\hat{S}_{c})$ is bounded a.s. following
(143). Therefore, we again have $E(|Q_{12}|)=o(1)$ from Assumption 6, so
$Q_{12}=o_{p}(1)$ by Markov’s inequality.

$Q_{13}=o_{p}(1)$ can be shown using exactly the same arguments. We therefore
have $Q_{1}=o_{p}(1)$.

Next, we show $Q_{2}=o_{p}(1)$. First, we write

$\displaystyle Q_{2}=Q_{21}-Q_{22}-Q_{23}$ (160)

where

$\displaystyle Q_{21}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\pi^{*},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(1)}(t;\pi^{*},S^{*},\hat{S}_{c})$
(161) $\displaystyle Q_{22}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\left\\{\bar{A}(t;\beta,\pi^{*},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta,\pi^{*},S^{*},\hat{S}_{c})\right\\}d\mathcal{N}_{i}^{(0)}(t;\pi^{*},\hat{S},\hat{S}_{c})$
(162) $\displaystyle Q_{23}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},S^{*},\hat{S}_{c})\left\\{d\mathcal{N}_{i}^{(0)}(t;\pi^{*},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\pi^{*},S^{*},\hat{S}_{c})\right\\}.$
(163)

Consider $Q_{21}$. We have

$\displaystyle Q_{21}=Q_{211}-Q_{212}-Q_{213},$ (164)

where

$\displaystyle Q_{211}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\frac{A_{i}\\{\hat{S}(\tau;A_{i},Z_{i})-S^{*}(\tau;A_{i},Z_{i})\\}}{\pi^{*}(Z_{i})^{A_{i}}\\{1-\pi^{*}(Z_{i})\\}^{1-A_{i}}}+\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}a\\{\hat{S}(\tau;a,Z_{i})-S^{*}(\tau;a,Z_{i})\\},$
$\displaystyle Q_{212}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}\frac{aA_{i}^{a}(1-A_{i})^{1-a}}{\pi^{*}(Z_{i})^{a}\\{1-\pi^{*}(Z_{i})\\}^{1-a}}\int_{0}^{\tau}J_{i}(t;a,\hat{S},\hat{S}_{c})\\{d\hat{S}(t;a,Z_{i})-dS^{*}(t;a,Z_{i})\\},$
$\displaystyle Q_{213}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}\frac{aA_{i}^{a}(1-A_{i})^{1-a}}{\pi^{*}(Z_{i})^{a}\\{1-\pi^{*}(Z_{i})\\}^{1-a}}$
$\displaystyle\times\int_{0}^{\tau}\left[\int_{0}^{t}\left\\{\frac{1}{\hat{S}(u;a,Z_{i})}-\frac{1}{S^{*}(u;a,Z_{i})}\right\\}\frac{dM_{ci}(u;a,\hat{S}_{c})}{\hat{S}_{c}(u;a,Z_{i})}\right]dS^{*}(t;a,Z_{i}).$

For $Q_{211}$, we can easily see that

$\displaystyle|Q_{211}|\lesssim\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}|\\{\hat{S}(\tau;a,Z_{i})-S^{*}(\tau;a,Z_{i})\\}|\lesssim\frac{1}{n}\sum_{i=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}\right|,$
(165)

so $E(|Q_{211}|)=o(1)$ by Assumption 6 and $Q_{211}=o_{p}(1)$ by Markov’s
inequality.

Term $Q_{212}$ involves a difference in increments
$d\hat{S}(t;a,Z_{i})-dS^{*}(t;a,Z_{i})$. Applying integration by parts to the
integral term we have

$\displaystyle\int_{0}^{\tau}J_{i}(t;a,\hat{S},\hat{S}_{c})\\{d\hat{S}(t;a,Z_{i})-dS^{*}(t;a,Z_{i})\\}$
(166) $\displaystyle=$
$\displaystyle\left[J_{i}(t;a,\hat{S},\hat{S}_{c})\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}\right]\bigg{|}_{0}^{\tau}-\int_{0}^{\tau}\frac{\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}dM_{ci}(t;a,\hat{S}_{c})}{\hat{S}(t;a,Z_{i})\hat{S}_{c}(t;a,Z_{i})},$
(167)

So

$\displaystyle Q_{212}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}\frac{aA_{i}^{a}(1-A_{i})^{1-a}}{\pi^{*}(Z_{i})^{a}\\{1-\pi^{*}(Z_{i})\\}^{1-a}}\left[J_{i}(t;a,\hat{S},\hat{S}_{c})\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}\right]\bigg{|}_{0}^{\tau}$
(168)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\sum_{a=0,1}\frac{aA_{i}^{a}(1-A_{i})^{1-a}}{\pi^{*}(Z_{i})^{a}\\{1-\pi^{*}(Z_{i})\\}^{1-a}}\int_{0}^{\tau}\frac{\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}dM_{ci}(t;a,\hat{S}_{c})}{\hat{S}(t;a,Z_{i})\hat{S}_{c}(t;a,Z_{i})}.$
(169)

Note that
$dM_{ci}(t;a,\hat{S}_{c})=dN_{ci}(t)-Y_{i}(t)d\hat{\Lambda}_{c}(t;a,Z_{i})$.
Since both $N_{ci}(t)$ and $\hat{\Lambda}_{c}(t;a,Z_{i})$ are monotone
functions, we may again apply (132) on the second term above. The nuisance
functions are bounded away from zero a.s., so we have

$\displaystyle|Q_{212}|\lesssim$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\\{\hat{S}(t;a,Z_{i})-S^{*}(t;a,Z_{i})\\}\right|,$
(170)

so $E(|Q_{212}|)=o(1)$ from Assumption 6 and $Q_{212}=o_{p}(1)$ by Markov’s
inequality.

By applying (132) twice on each of the double integrals in $Q_{213}$, we can
show $Q_{213}=o_{p}(1)$ in exactly the same way.

Same approach used for $Q_{21}$ also gives $Q_{22}=o_{p}(1)$ and
$Q_{23}=o_{p}(1)$. We hence have $Q_{2}=o_{p}(1)$.

$Q_{3}=o_{p}(1)$ can again be shown using the same techniques we use for
$Q_{2}$, so we omit the details.

Lastly, we show that
$U(\beta,\pi^{*},S^{*},S_{c}^{*})=\mu(\beta,\pi^{*},S^{*},S_{c}^{*})+o_{p}(1)$
for $\beta\in\mathcal{B}$.

From the definition of the AIPW estimating functions
$D_{1i}(t;\beta^{*},\Lambda^{*},\pi,S,S_{c})$ and
$D_{2i}(\beta^{*},\Lambda^{*},\pi,S,S_{c})$, we have

$\displaystyle
d\mathcal{N}_{i}^{(0)}(t;\pi,S,S_{c})=D_{1i}(t;\beta^{*},\Lambda^{*},\pi,S,S_{c})+\Gamma_{i}^{(0)}(t;\beta^{*},\pi,S,S_{c})d\Lambda^{*}(t),$
$\displaystyle\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\pi,S,S_{c})=D_{2i}(\beta^{*},\Lambda^{*},\pi,S,S_{c})+\int_{0}^{\tau}\Gamma_{i}^{(1)}(t;\beta^{*},\pi,S,S_{c})d\Lambda^{*}(t).$

For $\beta\in\mathcal{B}$, we apply this to $U(\beta,\pi^{*},S^{*},S_{c}^{*})$
and have

$\displaystyle U(\beta,\pi^{*},S^{*},S_{c}^{*})$ (171) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})d\mathcal{N}_{i}^{(0)}(t;\pi^{*},S^{*},S_{c}^{*})$
(172) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\bigg{[}D_{2i}(\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})+\int_{0}^{\tau}\Gamma_{i}^{(1)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})d\Lambda^{*}(t)$
(173)
$\displaystyle-\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\\{\Gamma_{i}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})d\Lambda^{*}(t)+D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})\\}\bigg{]}$
(174) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\frac{1}{n}\sum_{i=1}^{n}\\{\Gamma_{i}^{(1)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\Gamma_{i}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\\}d\Lambda^{*}(t)$
(175)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}D_{2i}(\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(176)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(177) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\\{\mathcal{S}^{(1)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\\}d\Lambda^{*}(t)$
(178)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}D_{2i}(\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(179)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(180) $\displaystyle=$
$\displaystyle\int_{0}^{\tau}\left\\{\bar{A}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\right\\}\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})d\Lambda^{*}(t)$
(181)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}D_{2i}(\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(182)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*}).$
(183)

Next, for each of $\bar{A}$ and $\mathcal{S}^{(0)}$, we add and subtract its
limits and have

$\displaystyle U(\beta,\pi^{*},S^{*},S_{c}^{*})$ (184) $\displaystyle=$
$\displaystyle\quad\mu(\beta,\pi^{*},S^{*},S_{c}^{*})$ (185)
$\displaystyle+\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})-\bar{\alpha}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})+\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\\}$
(186)
$\displaystyle\quad\times\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})d\Lambda^{*}(t)$
(187)
$\displaystyle+\int_{0}^{\tau}\\{\bar{\alpha}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\\}$
(188)
$\displaystyle\quad\times\\{\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\mathpzc{s}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\\}d\Lambda^{*}(t)$
(189)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}D_{2i}(\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(190)
$\displaystyle-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$
(191)
$\displaystyle+\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\\}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*}).$
(192)

For (187), since $\Lambda^{*}(t)$ is an increasing function and
$\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})$ is bounded a.s., we
can apply (132) to it and have

$\displaystyle\bigg{|}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})-\bar{\alpha}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})+\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\\}$
(193)
$\displaystyle\quad\times\mathcal{S}^{(0)}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})d\Lambda^{*}(t)\bigg{|}$
(194) $\displaystyle\lesssim$
$\displaystyle\sup_{t\in[0,\tau]}\left|\bar{A}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\bar{A}(t;\beta,\pi^{*},S^{*},S_{c}^{*})-\bar{\alpha}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})+\bar{\alpha}(t;\beta,\pi^{*},S^{*},S_{c}^{*})\right|,$
(195)

which is $o_{p}(1)$ from Assumption 8. Similarly, (189) is $o_{p}(1)$.

Next, we note that the increments in
$D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$ are $dN_{i}(t)$,
$dS^{*}(t;A_{i},Z_{i})$, $dS^{*}(t;a,Z_{i})$ and $d\Lambda(t)$, all of which
are increments of monotone functions. So similar to (187) and (189), we can
apply (132), the strict positivity Assumptions 4 and Assumption 8 to show that
(192) is $o_{p}(1)$.

Since we have $S^{*}=S^{o}$ or $(\pi^{*},S_{c}^{*})=(\pi^{o},S_{c}^{o})$,
Theorem 1′ gives that both (190) and (191) are sums of i.i.d. mean zero terms.
The strict positivity Assumption 4 ensures that these i.i.d. mean zero terms
are also bounded, hence having bounded variance. So
$\eqref{AIPW:l2.7}=o_{p}(1)$ and $\eqref{AIPW:l2.8}=o_{p}(1)$ by the weak law
of large numbers.

The second part of the Lemma,

$\displaystyle\frac{\partial}{\partial\beta}U(\beta,\hat{\pi},\hat{S},\hat{S}_{c})\overset{p}{\to}-\nu(\beta,\pi^{*},S^{*},S_{c}^{*}),$
(196)

can be shown using exactly the same arguments as how we proved
$U(\beta,\hat{\pi},\hat{S},\hat{S}_{c})\overset{p}{\to}\mu(\beta,\pi^{*},S^{*},S_{c}^{*})$
above, which completes the proof.

∎

Proof of Lemma 8 First, write

$\displaystyle\sqrt{n}U(\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})=$
$\displaystyle\sqrt{n}U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})+Q_{4}+Q_{5}+Q_{6},$
(197)

where

$\displaystyle Q_{4}=$
$\displaystyle\sqrt{n}\\{U(\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})-U(\beta^{*},\pi^{o},\hat{S},S_{c}^{o})\\}-\sqrt{n}\\{U(\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-U(\beta^{*},\pi^{o}S^{o},S_{c}^{o})\\},$
(198) $\displaystyle Q_{5}=$
$\displaystyle\sqrt{n}\\{U(\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\},$
(199) $\displaystyle Q_{6}=$
$\displaystyle\sqrt{n}\\{U(\beta^{*},\pi^{o},\hat{S},S_{c}^{o})-U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}.$
(200)

The structure of the proof is as follows: we first show that using the rate
condition Assumption 7 among other assumptions that $Q_{4}$, which is a
difference in differences, is $o_{p}(1)$. Next, we show that $Q_{5}$ and
$Q_{6}$ are $o_{p}(1)$, which uses, among other assumptions, the independence
between in-fold and out-of-fold data induced by cross-fitting. Finally, we
show that $\sqrt{n}U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})$ is asymptotically
equivalent to a sum of i.i.d. terms.

We first show that $Q_{4}=o_{p}(1)$. For any fixed nuisance function $S$, we
have

$\displaystyle\sqrt{n}\\{U_{1}(\beta^{*},\hat{\pi},S,\hat{S}_{c})-U_{1}(\beta^{*},\pi^{o},S,S_{c}^{o})\\}$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\hat{\pi},S,\hat{S}_{c})-d\mathcal{N}_{i}^{(1)}(t;\pi^{o},S,S_{c}^{o})$
$\displaystyle-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta^{*},\pi^{o},S,S_{c}^{o})\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S,\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\pi^{o},S,S_{c}^{o})\\}$
$\displaystyle-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S,\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S,S_{c}^{o})\\}d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S,\hat{S}_{c}).$

So we can write

$\displaystyle Q_{4}=Q_{41}-Q_{42}-Q_{43}-Q_{44},$ (201)

where

$\displaystyle Q_{41}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(1)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(1)}(t;\pi^{o},\hat{S},S_{c}^{o})-d\mathcal{N}_{i}^{(1)}(t;\hat{\pi},S^{o},\hat{S}_{c})+d\mathcal{N}_{i}^{(1)}(t;\pi^{o},S^{o},S_{c}^{o})$
(202)
$\displaystyle-\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})$
(203)
$\displaystyle\quad\times\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\pi^{o},\hat{S},S_{c}^{o})-d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S^{o},\hat{S}_{c})+d\mathcal{N}_{i}^{(0)}(t;\pi^{o},S^{o},S_{c}^{o})\\}$
(204) $\displaystyle Q_{42}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S^{o},\hat{S}_{c})\\}$
(205) $\displaystyle Q_{43}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{o},\hat{S},S_{c}^{o})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\pi^{o},\hat{S},S_{c}^{o})\\}$
(206) $\displaystyle Q_{44}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})$
(207)
$\displaystyle\times\\{\bar{A}(t;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},\hat{S},S_{c}^{o})-\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})+\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}.$
(208)

Consider $Q_{41}$, which can be written as
$Q_{41}=-Q_{411}+Q_{412}-Q_{413}-Q_{414}$, where

$\displaystyle Q_{411}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sum_{a=0,1}\frac{A_{i}^{a}(1-A_{i})^{1-a}}{\pi^{o}(Z_{i})^{a}\\{1-\pi^{o}(Z_{i})\\}^{1-a}}\int_{0}^{\tau}\Bigg{[}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}$
(209)
$\displaystyle\times\int_{0}^{t}\left\\{\frac{d\hat{S}(t;a,Z_{i})}{\hat{S}(u;a,Z_{i})}-\frac{dS^{o}(t;a,Z_{i})}{S^{o}(u;a,Z_{i})}\right\\}\left\\{\frac{dM_{ci}(u;a,\hat{S}_{c})}{\hat{S}_{c}(u;a,Z_{i})}-\frac{dM_{ci}(u;a,S_{c}^{o})}{S_{c}^{o}(u;a,Z_{i})}\right\\}\Bigg{]}$
(210) $\displaystyle Q_{412}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\\{A_{i}-\bar{A}(\tau;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\left\\{\frac{1}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}-\frac{1}{\pi^{o}(Z_{i})^{A_{i}}\\{1-\pi^{o}(Z_{i})\\}^{1-A_{i}}}\right\\}$
(211)
$\displaystyle\times\left\\{\hat{S}(\tau;A_{i},Z_{i})-S^{o}(\tau;A_{i},Z_{i})\right\\}$
(212) $\displaystyle Q_{413}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sum_{a=0,1}A_{i}^{a}(1-A_{i})^{1-a}\left\\{\frac{1}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}-\frac{1}{\pi^{o}(Z_{i})^{a}\\{1-\pi^{o}(Z_{i})\\}^{1-a}}\right\\}$
(213)
$\displaystyle\times\int_{0}^{\tau}dS^{o}(t;a,Z_{i})\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\int_{0}^{t}\left\\{\frac{1}{\hat{S}(u;a,Z_{i})}-\frac{1}{S^{o}(u;a,Z_{i})}\right\\}\frac{dM_{ci}(u;a,\hat{S}_{c})}{\hat{S}_{c}(u;a,Z_{i})}$
(214) $\displaystyle Q_{414}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sum_{a=0,1}A_{i}^{a}(1-A_{i})^{1-a}\left\\{\frac{1}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}-\frac{1}{\pi^{o}(Z_{i})^{a}\\{1-\pi^{o}(Z_{i})\\}^{1-a}}\right\\}$
(215)
$\displaystyle\times\int_{0}^{\tau}\\{d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(t;a,\hat{S},\hat{S}_{c}).$
(216)

For $Q_{411}$, we first notice that by the strict positivity Assumption 4,
$A_{i}^{a}(1-A_{i})^{1-a}/\\{\pi^{o}(Z_{i})^{a}\\{1-\pi^{o}(Z_{i})\\}^{1-a}\\}$
is bounded a.s.. The expectation of the absolute value of the double integral
in $Q_{411}$ can be bounded directly using $\mathcal{D}^{\dagger}_{1}$ defined
in Assumption 7, which leads to

$\displaystyle E(|Q_{411}|)\lesssim$
$\displaystyle\sqrt{n}\mathcal{D}^{\dagger}_{1}=o(1),$ (217)

where the last equality follows from rate condition Assumption 7.

As discussed in the Asymptotic Properties Section, integral remainders
$\mathcal{D}^{\dagger}_{1}$ is specific to our case because both nuisance
functions $S(t;a,z)$ and $S_{c}(t;a,z)$ are time-dependent, which can lead to
a product between the differences $\hat{S}_{c}-S_{c}$ and differences of
increments $d\hat{S}-dS^{o}$, like in (210). To the best of our knowledge,
remainder terms like this can not be sufficiently controlled using existing
tools, which requires us to make additional assumptions, such as
$\mathcal{D}_{1}^{\dagger}(\hat{S},\hat{S}_{c};S^{o},S_{c}^{o})=o(n^{-1/2})$
in the rate condition Assumption 7.

For $Q_{412}$, recall that
$A_{i}-\bar{A}(\tau;\beta^{*},\pi^{o},S^{o},S_{c}^{o})$ is bounded a.s. and
$\hat{\pi}(Z_{i})$, $\pi^{o}(Z_{i})$, $1-\hat{\pi}(Z_{i})$ and
$1-\pi^{o}(Z_{i})$ are bounded away from zero a.s., so we have

$\displaystyle|Q_{412}|\leq$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|A_{i}-\bar{A}(\tau;\beta^{*},\pi^{o},S^{o},S_{c}^{o})|\cdot\frac{|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|\cdot\left|\hat{S}(\tau;A_{i},Z_{i})-S^{o}(\tau;A_{i},Z_{i})\right|}{|\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}\pi^{o}(Z_{i})^{A_{i}}\\{1-\pi^{o}(Z_{i})\\}^{1-A_{i}}|}$
(218) $\displaystyle\lesssim$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|.$
(219)

Therefore

$\displaystyle E(|Q_{412}|)\lesssim$
$\displaystyle\sqrt{n}E^{\dagger}\Bigg{\\{}E\Bigg{[}|\hat{\pi}(Z)-\pi^{o}(Z)|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z)-S^{o}(t;a,Z)\right|\Bigg{]}\Bigg{\\}}$
(220) $\displaystyle\leq$
$\displaystyle\sqrt{n}\left\|\hat{\pi}-\pi^{o}\right\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}$
(221) $\displaystyle=$ $\displaystyle o(1),$ (222)

where (221) follows from the Cauchy-Schwartz inequality $|E(AB)|^{2}\leq
E(A^{2})E(B^{2})$, while (222) uses the rate condition Assumption 7.

$Q_{413}$ can be bounded similarly with the help of (132). First we note that
$S^{o}(t;a,Z_{i})$ is a monotone function by assumption. Recall that
$dM_{ci}(u;a,\hat{S}_{c})=dN_{ci}(u)-Y_{i}(u)d\hat{\Lambda}_{c}(u;a,Z_{i})$ is
also a sum of two terms, each being the product of a term bounded a.s. and an
increment of a monotone function. We therefore apply (132) twice to each of
the double integral in $Q_{413}$ and have

$\displaystyle|Q_{413}|\lesssim$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\sum_{a=0,1}\frac{|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|}{|\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}\pi^{o}(Z_{i})^{a}\\{1-\pi^{o}(Z_{i})\\}^{1-a}|}$
(223)
$\displaystyle\times\sup_{t\in[0,\tau]}\left|\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\int_{0}^{t}\left\\{\frac{1}{\hat{S}(u;a,Z_{i})}-\frac{1}{S^{o}(u;a,Z_{i})}\right\\}\frac{dM_{ci}(u;a,\hat{S}_{c})}{\hat{S}_{c}(u;a,Z_{i})}\right|$
(224) $\displaystyle\lesssim$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|$
(225)
$\displaystyle\times\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left\\{\sup_{u\in[0,t]}\left|\frac{\hat{S}(u;a,Z_{i})-S^{o}(u;a,Z_{i})}{\hat{S}(u;a,Z_{i})S^{o}(u;a,Z_{i})\hat{S}_{c}(u;a,Z_{i})}\right|\right\\}$
(226) $\displaystyle\lesssim$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|.$
(227)

So we again have

$\displaystyle E(|Q_{413}|)\lesssim$
$\displaystyle\sqrt{n}E^{\dagger}\Bigg{\\{}E\Bigg{[}|\hat{\pi}(Z)-\pi^{o}(Z)|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z)-S^{o}(t;a,Z)\right|\Bigg{]}\Bigg{\\}}$
(228) $\displaystyle\leq$
$\displaystyle\sqrt{n}\left\|\hat{\pi}-\pi^{o}\right\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}$
(229) $\displaystyle=$ $\displaystyle o(1)$ (230)

from the Cauchy-Schwartz inequality and the rate condition Assumption 7.

The integral in $Q_{414}$ involves a difference in increments
$d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})$, so we apply integration by parts and
have

$\displaystyle\int_{0}^{\tau}\\{d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})$
(231) $\displaystyle=$
$\displaystyle\left[\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})\right]\bigg{|}_{0}^{\tau}$
(232)
$\displaystyle-\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}\frac{\partial}{\partial
t}\left[\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})\right]$
(233) $\displaystyle=$
$\displaystyle\\{\hat{S}(\tau;a,Z_{i})-S^{o}(\tau;a,Z_{i})\\}\\{a-\bar{A}(\tau;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(\tau;a,\hat{S},\hat{S}_{c})$
(234)
$\displaystyle-\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\cdot\frac{dM_{ci}(t;a,\hat{S}_{c})}{\hat{S}(t;a,Z_{i})\hat{S}_{c}(t;a,Z_{i})}$
(235)
$\displaystyle+\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}\frac{\partial}{\partial
t}\left[\frac{\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})}{\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})}\right]\cdot
J_{i}(t;a,\hat{S},\hat{S}_{c})$ (236) $\displaystyle=$
$\displaystyle\\{\hat{S}(\tau;a,Z_{i})-S^{o}(\tau;a,Z_{i})\\}\\{a-\bar{A}(\tau;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(\tau;a,\hat{S},\hat{S}_{c})$
(237)
$\displaystyle\\-\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\cdot\frac{dM_{ci}(t;a,\hat{S}_{c})}{\hat{S}(t;a,Z_{i})\hat{S}_{c}(t;a,Z_{i})}$
(238)
$\displaystyle+\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})\frac{1}{\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})}\cdot\frac{1}{n}\sum_{j=1}^{n}d\Gamma_{j}^{(1)}(t;\beta^{*},\pi^{o},S^{o},S^{o}_{c})$
(239)
$\displaystyle-\int_{0}^{\tau}\\{\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})\frac{\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})}{\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})^{2}}\cdot\frac{1}{n}\sum_{j=1}^{n}d\Gamma_{j}^{(0)}(t;\beta^{*},\pi^{o},S^{o},S^{o}_{c}),$
(240)

where the last two equalities follow from the product rule. For $l=0,1$, we
again apply the product rule and have

$\displaystyle d\Gamma_{j}^{(l)}(t;\beta^{*},\pi^{o},S^{o},S^{o}_{c})$ (241)
$\displaystyle=$
$\displaystyle\frac{A^{l}_{j}e^{\beta^{*}A_{j}}}{\pi^{o}(Z_{j})^{A_{j}}\\{1-\pi^{o}(Z_{j})\\}^{1-A_{j}}S^{o}_{c}(t;A_{j},Z_{j})}dY_{j}(t)$
(242)
$\displaystyle-\frac{A^{l}_{j}Y_{j}(t)e^{\beta^{*}A_{j}}}{\pi^{o}(Z_{j})^{A_{j}}\\{1-\pi^{o}(Z_{j})\\}^{1-A_{j}}S^{o}_{c}(t;A_{j},Z_{j})^{2}}dS^{o}_{c}(t;A_{j},Z_{j})-\frac{A^{l}_{j}e^{\beta^{*}A_{j}}}{\pi^{o}(Z_{j})^{A_{j}}\\{1-\pi^{o}(Z_{j})\\}^{1-A_{j}}}dS^{o}(t;A_{j},Z_{j})$
(243)
$\displaystyle+\sum_{a=0,1}a^{l}\left\\{1+\frac{A_{j}^{a}(1-A_{j})^{1-a}}{\pi^{o}(Z_{j})^{a}\\{1-\pi^{o}(Z_{j})\\}^{1-a}}J_{j}(t;a,S^{o},S^{o}_{c})\right\\}e^{\beta^{*}a}dS^{o}(t;a,Z_{j})$
(244)
$\displaystyle+\sum_{a=0,1}a^{l}\frac{A_{j}^{a}(1-A_{j})^{1-a}}{\pi^{o}(Z_{j})^{a}\\{1-\pi^{o}(Z_{j})\\}^{1-a}}\frac{S^{o}(t;a,Z_{j})e^{\beta^{*}a}}{S^{o}(t;a,Z_{j})S^{o}_{c}(t;a,Z_{j})}dM_{cj}(u;a,S^{o}_{c}).$
(245)

Since $dM_{cj}(u;a,S^{o}_{c})=dN_{cj}(u)-Y_{j}(u)d\Lambda_{c}^{o}(u;a,Z_{j})$,
we can now see that $d\Gamma_{j}^{(l)}(t;\beta^{*},\pi^{o},S^{o},S^{o}_{c})$
is once again a sum of terms, each being a product between a term that is
bounded a.s. and an increment of a monotone function. Therefore, applying
(132), we have

$\displaystyle\left|\int_{0}^{\tau}\\{d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})\\}\\{a-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}J_{i}(t;a,\hat{S},\hat{S}_{c})\right|$
(246) $\displaystyle\lesssim$
$\displaystyle|\hat{S}(\tau;a,Z_{i})-S^{o}(\tau;a,Z_{i})|$ (247)
$\displaystyle+\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|$
(248)
$\displaystyle+\frac{1}{n}\sum_{j=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|$
(249)
$\displaystyle+\frac{1}{n}\sum_{j=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|$
(250) $\displaystyle\lesssim$
$\displaystyle\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|.$
(251)

So

$\displaystyle|Q_{414}|\lesssim\frac{1}{\sqrt{n}}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z_{i})-S^{o}(t;a,Z_{i})\right|,$
(252)

and we again have

$\displaystyle E(|Q_{414}|)\lesssim$
$\displaystyle\sqrt{n}E^{\dagger}\Bigg{\\{}E\Bigg{[}|\hat{\pi}(Z)-\pi^{o}(Z)|\cdot\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|\hat{S}(t;a,Z)-S^{o}(t;a,Z)\right|\Bigg{]}\Bigg{\\}}$
(253) $\displaystyle\leq$
$\displaystyle\sqrt{n}\left\|\hat{\pi}-\pi^{o}\right\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}$
(254) $\displaystyle=$ $\displaystyle o(1),$ (255)

from the Cauchy-Schwartz inequality and the rate condition Assumption 7.

Therefore, we have

$\displaystyle E(|Q_{41}|)\leq
E(|Q_{411}|)+E(|Q_{412}|)+E(|Q_{413}|)+E(|Q_{414}|)\lesssim o(1),$ (256)

so $Q_{41}=o_{p}(1)$ by Markov’s inequality.

Next, we bound $Q_{42}$, which involves the use of
$\mathcal{D}^{\dagger}_{2}$. First, we let $Q_{42}=Q_{421}+Q_{422}$, where

$\displaystyle Q_{421}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\}\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S^{o},\hat{S}_{c})\\}$
(257) $\displaystyle Q_{422}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\\{d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S^{o},\hat{S}_{c})\\}.$
(258)

Note that like how we bounded $|Q_{12}|$ earlier, we also have

$\displaystyle\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})$
(259) $\displaystyle=$
$\displaystyle\frac{\mathcal{S}^{(1)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})-\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})}{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}$
(260) $\displaystyle=$
$\displaystyle\frac{\\{\mathcal{S}^{(1)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\}\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}$
(261)
$\displaystyle-\frac{\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\}}{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}$
(262) $\displaystyle=$
$\displaystyle\frac{\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}\cdot\frac{1}{n}\sum_{j=1}^{n}\\{\Gamma^{(1)}_{j}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\Gamma^{(1)}_{j}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\}$
(263)
$\displaystyle-\frac{\mathcal{S}^{(1)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}{\mathcal{S}^{(0)}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\mathcal{S}^{(0)}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})}\cdot\frac{1}{n}\sum_{j=1}^{n}\\{\Gamma^{(0)}_{j}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\Gamma^{(0)}_{j}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})\\}$
(264) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{j=1}^{n}C_{j}(t)\\{\hat{\pi}(Z_{j})-\pi^{o}(Z_{j})\\},$
(265)

where $C_{j}(t)$ are some functions bounded a.s.. Similarly, we have

$\displaystyle\\{\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}=\frac{1}{n}\sum_{j=1}^{n}C_{j}^{\prime}(t)\\{\hat{S}_{c}(t;a,Z_{i})-S^{o}_{c}(t;a,Z_{i})\\},$
(266)

where $C_{j}^{\prime}(t)$ are some other functions bounded a.s..

Next, let
$d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}_{i}^{(0)}(t;\hat{\pi},S^{o},\hat{S}_{c})=K_{1i}+K_{2i}$,
where

$\displaystyle K_{1i}=$
$\displaystyle\frac{d\hat{S}(t;A_{i},Z_{i})-dS^{o}(t;A_{i},Z_{i})}{\hat{\pi}(Z_{i})^{A_{i}}\\{1-\hat{\pi}(Z_{i})\\}^{1-A_{i}}}$
$\displaystyle-\sum_{a=0,1}\left\\{1+\frac{A_{i}^{a}(1-A_{i})^{1-a}}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}J_{i}(t;a,S^{o},\hat{S}_{c})\right\\}\\{d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})\\}$
$\displaystyle K_{2i}=$
$\displaystyle-\sum_{a=0,1}\frac{A_{i}^{a}(1-A_{i})^{1-a}}{\hat{\pi}(Z_{i})^{a}\\{1-\hat{\pi}(Z_{i})\\}^{1-a}}\int_{0}^{t}\left\\{\frac{1}{\hat{S}(u;a,Z_{i})}-\frac{1}{S^{o}(u;a,Z_{i})}\right\\}\frac{dM_{ci}(u;a,Z_{i})}{\hat{S}_{c}(u;a,Z_{i})}\cdot
d\hat{S}(t;a,Z_{i}).$

We now have $Q_{421}=Q_{4211}+Q_{4212}$, where

$\displaystyle Q_{4211}=$
$\displaystyle\frac{1}{n^{3/2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\\{\hat{\pi}(Z_{j})-\pi^{o}(Z_{j})\\}\int_{0}^{\tau}C_{j}(t)K_{1i}$
(267) $\displaystyle Q_{4212}=$
$\displaystyle\frac{1}{n^{3/2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\\{\hat{\pi}(Z_{j})-\pi^{o}(Z_{j})\\}\int_{0}^{\tau}C_{j}(t)K_{2i}.$
(268)

For $Q_{4212}$, we can apply (132), the rate Condition Assumption 7 and the
boundedness of appropriate terms to show that

$\displaystyle
E(|Q_{4212}|)\lesssim\sqrt{n}\|\hat{\pi}-\pi^{o}\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}=o(1).$
(269)

$\int_{0}^{\tau}C_{j}(t)K_{1i}$ in $Q_{4211}$ involves stochastic differences
$d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})$, so like in (167) and (240) we first
apply integration by parts to turn $d\hat{S}-dS^{o}$ into $\hat{S}-S^{o}$.
Like (241), the $dC_{j}(t)$ term we have as a result of integration by parts
can again be shown to be a sum of terms, each being a product between a term
that is bounded a.s. and an increment of a monotone function. This allows us
to apply (132), the rate Condition Assumption 7 and the boundedness of
appropriate terms, which leads to
$E(|Q_{4211}|)\lesssim\sqrt{n}\|\hat{\pi}-\pi^{o}\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}=o(1)$.
We therefore have $E(|Q_{421}|)=o(1)$ and $Q_{421}=o_{p}(1)$ by Markov’s
inequality.

For term $Q_{422}$, we first write $Q_{422}=Q_{4221}+Q_{4222}$, where

$\displaystyle Q_{4221}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}K_{1i},$
(270) $\displaystyle Q_{4222}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{o},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}K_{2i}.$
(271)

$E(|Q_{4221}|)$ involves a product between
$d\hat{S}(t;a,Z_{i})-dS^{o}(t;a,Z_{i})$ and
$\hat{S}_{c}(t;a,Z_{i})-S_{c}^{o}(t;a,Z_{i})$, which can not be bounded using
any existing tools we have. Therefore, we directly bound it using
$\mathcal{D}^{\dagger}_{2}$ in Assumption 7, which gives

$\displaystyle E(|Q_{4221}|)\lesssim\sqrt{n}\mathcal{D}^{\dagger}_{2}=o(1).$
(272)

Next, using (266), we have

$\displaystyle Q_{4222}=$
$\displaystyle\frac{1}{n^{3/2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\int_{0}^{\tau}C_{j}^{\prime}(t)\\{\hat{S}_{c}(t;a,Z_{i})-S^{o}_{c}(t;a,Z_{i})\\}K_{2i}.$
(273)

By again applying (132), the rate condition Assumption 7 and the boundedness
of appropriate terms to $Q_{4222}$, we have

$\displaystyle
E(|Q_{4222}|)\lesssim\sqrt{n}\left\|\hat{S}_{c}-S_{c}^{o}\right\|_{\dagger}\cdot\left\|\hat{S}-S^{o}\right\|_{\dagger}=o(1).$
(274)

Therefore, $E(|Q_{422}|)=o(1)$ from rate condition Assumption 7.

Combining our results, we have

$\displaystyle E(|Q_{42}|)\leq E(|Q_{421}|)+E(|Q_{422}|)=o(1).$ (275)

Using the same techniques we used for $Q_{41}$ and $Q_{42}$ above, with the
rate condition Assumption 7 and without using $\mathcal{D}^{\dagger}$, we can
show that $E(|Q_{43}|)=o(1)$ and $E(|Q_{44}|)=o(1)$.

Hence we conclude that $E(|Q_{4}|)\leq$
$E(|Q_{41}|)+E(|Q_{42}|)+E(|Q_{43}|)+E(|Q_{44}|)=o(1)$. Then by Markov’s
inequality, $Q_{4}=o_{p}(1)$.

Next, we show that $Q_{5}=o_{p}(1)$.

Using the definition of
$D_{1i}(\beta,\Lambda,\pi,S,S_{c}),D_{2i}(\beta,\Lambda,\pi,S,S_{c})$ defined
in Supplementary Material A, it can be verified that

$\displaystyle
U(\beta^{*},\pi,S,S_{c})=\frac{1}{n}\sum_{i=1}^{n}\left[D_{2i}(\beta^{*},\Lambda^{*},\pi,S,S_{c})-\int_{0}^{\tau}\bar{A}(t;\beta^{*},\pi,S,S_{c})D_{1i}(t;\beta^{*},\Lambda^{*},\pi,S,S_{c})\right].$
(276)

So we have $Q_{5}=Q_{51}-Q_{52}-Q_{53}-Q_{54}$, where

$\displaystyle Q_{51}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\\{D_{2i}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2i}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})\\}$
(277) $\displaystyle Q_{52}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\\{D_{1i}(t;\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})\\}$
(278) $\displaystyle Q_{53}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\\}$
(279)
$\displaystyle\times\\{D_{1i}(t;\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})\\}$
(280) $\displaystyle Q_{54}=$
$\displaystyle\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o}).$
(281)

First, consider $Q_{51}$. By the law of total variance, we have

$\displaystyle\mbox{Var}(Q_{51})=\mbox{Var}\\{E(Q_{51}|O^{\dagger})\\}+E\\{\mbox{Var}(Q_{51}|O^{\dagger})\\}.$
(282)

We note from Theorem 1′ that
$E\\{D_{2i}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2i}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})|O^{\dagger}\\}=0$
for each $i$, where $O^{\dagger}$ is the sample independent from $O$ that is
used for estimating the nuisance functions, so $E(Q_{51}|O^{\dagger})=0$.
Moreover, when conditional on $O^{\dagger}$, $Q_{51}$ is a sample average of
mean-zero i.i.d terms, so we have

$\displaystyle\mbox{Var}(Q_{51}|O^{\dagger})=$
$\displaystyle\frac{n}{n}E\left[\\{D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})\\}^{2}|O^{\dagger}\right].$
(283)

Expand
$D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$,
we have

$\displaystyle
D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$
(284) $\displaystyle=$
$\displaystyle-\int_{0}^{\tau}\frac{A\\{\hat{\pi}(Z)-\pi^{o}(Z)\\}}{\hat{\pi}(Z)\pi^{o}(Z)}\cdot\\{dS^{o}(t;A,Z)+S^{o}(t;A,Z)e^{\beta^{*}A}d\Lambda^{*}(t)\\}$
(285)
$\displaystyle+\int_{0}^{\tau}\frac{AS_{c}^{o}(t;A,Z)\\{\pi^{o}(Z)-\hat{\pi}(Z)\\}}{\hat{\pi}(Z)S_{c}^{o}(t;A,Z)\hat{S}_{c}(t;A,Z)\pi^{o}(Z)^{A}\\{1-\pi^{o}(Z)\\}^{1-A}}\cdot\\{dN(t)-Y(t)e^{\beta^{*}}d\Lambda^{*}(t)\\}$
(286)
$\displaystyle+\int_{0}^{\tau}\frac{A\hat{\pi}(Z)\\{S_{c}^{o}(t;A,Z)-\hat{S}_{c}(t;A,Z)\\}}{\hat{\pi}(Z)S_{c}^{o}(t;A,Z)\hat{S}_{c}(t;A,Z)\pi^{o}(Z)^{A}\\{1-\pi^{o}(Z)\\}^{1-A}}\cdot\\{dN(t)-Y(t)e^{\beta^{*}}d\Lambda^{*}(t)\\}$
(287)
$\displaystyle-\int_{0}^{\tau}\frac{AJ(t;1,S^{o},S_{c}^{o})\\{\hat{\pi}(Z)-\pi^{o}(Z)\\}\\}}{\hat{\pi}(Z)\pi^{o}(Z)}\cdot\\{dS^{o}(t;1,Z)+S^{o}(t;1,Z)e^{\beta^{*}}d\Lambda^{*}(t)\\}$
(288)
$\displaystyle+\int_{0}^{\tau}\frac{A\\{J(t;1,S^{o},\hat{S}_{c})-J(t;1,S^{o},S_{c}^{o})\\}}{\hat{\pi}(Z)}\cdot\\{dS^{o}(t;1,Z)+S^{o}(t;1,Z)e^{\beta^{*}}d\Lambda^{*}(t)\\}.$
(289)

We now see that
$D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$
consists of several terms, where each term is an integral of a difference in
nuisance functions with respect to a monotone function. This allows us to
apply (132) to each of the terms and have

$\displaystyle|D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})|\lesssim|\hat{\pi}(Z)-\pi^{o}(Z)|+\sup_{t\in[0,\tau],a\in\\{0,1\\}}|S_{c}^{o}(t;a,Z)-\hat{S}_{c}(t;a,Z)|.$
(290)

From the inequality $(a+b)^{2}\leq 2a^{2}+2b^{2}$, we have

$\displaystyle\mbox{Var}(Q_{51}|O^{\dagger})\lesssim$ $\displaystyle
E\left[\left(|\hat{\pi}(Z)-\pi^{o}(Z)|+\sup_{t\in[0,\tau],a\in\\{0,1\\}}|S_{c}^{o}(t;a,Z)-\hat{S}_{c}(t;a,Z)|\right)^{2}\bigg{|}O^{\dagger}\right]$
(291) $\displaystyle\leq$ $\displaystyle
2E[\\{\hat{\pi}(Z)-\pi^{o}(Z)\\}^{2}|O^{\dagger}]+2E\left[\left\\{\sup_{t\in[0,\tau],a\in\\{0,1\\}}|S_{c}^{o}(t;a,Z)-\hat{S}_{c}(t;a,Z)|\right\\}^{2}\bigg{|}O^{\dagger}\right].$
(292)

So

$\displaystyle\mbox{Var}(Q_{51})$ (293) $\displaystyle=$
$\displaystyle\mbox{Var}^{\dagger}\\{E(Q_{51}|O^{\dagger})\\}+E^{\dagger}\\{\mbox{Var}(Q_{51}|O^{\dagger})\\}$
(294) $\displaystyle\lesssim$ $\displaystyle
0+E^{\dagger}(E[\\{\hat{\pi}(Z)-\pi^{o}(Z)\\}^{2}|O^{\dagger}])+E^{\dagger}\left(E\left[\left\\{\sup_{t\in[0,\tau],a\in\\{0,1\\}}|S_{c}^{o}(t;a,Z)-\hat{S}_{c}(t;a,Z)|\right\\}^{2}\bigg{|}O^{\dagger}\right]\right)$
(295) $\displaystyle=$
$\displaystyle\|\hat{\pi}-\pi^{o}\|_{\dagger}^{2}+\|\hat{S}_{c}-S_{c}^{o}\|_{\dagger}^{2}$
(296) $\displaystyle=$ $\displaystyle o(1).$ (297)

Therefore, $Q_{51}=o_{p}(1)$ by Chebyshev’s inequality.

Conditional on $O^{\dagger}$, we also have from Theorem 1′ that
$E\\{D_{1i}(t;\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})|O^{\dagger}\\}=0$
for each $t$ and $i$, so $Q_{52}$ is again a sample average of i.i.d. mean-
zero terms when conditional on $O^{\dagger}$, and we can show
$Q_{52}=o_{p}(1)$ in the same way as for $Q_{51}$ above.

Consider $Q_{53}$. Just like the expansion of
$D_{2}(\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{2}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$
in (284) above, we also have
$D_{1i}(t;\beta^{*},\Lambda^{*},\hat{\pi},S^{o},\hat{S}_{c})-D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$
as a sum of terms, where each term is a product between a difference in
nuisance functions and an increment of a monotone function. So same as in
$Q_{51}$, we apply (132) to each of the terms and have

$\displaystyle|Q_{53}|\lesssim$
$\displaystyle\sqrt{n}\sup_{t\in[0,\tau]}\left|\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\right|$
(298)
$\displaystyle\cdot\left\\{\frac{1}{n}\sum_{i=1}^{n}|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})|+\frac{1}{n}\sum_{i=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}|S_{c}^{o}(t;a,Z_{i})-\hat{S}_{c}(t;a,Z_{i})|\right\\}.$
(299)

From the uniform convergence Assumption 6 and the Markov’s inequality, we have

$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\left|\hat{\pi}(Z_{i})-\pi^{o}(Z_{i})\right|+\frac{1}{n}\sum_{i=1}^{n}\sup_{t\in[0,\tau],a\in\\{0,1\\}}\left|S_{c}^{o}(t;a,Z_{i})-\hat{S}_{c}(t;a,Z_{i})\right|=o_{p}(1).$
(300)

From (101) of Assumption 11, we have

$\displaystyle\sqrt{n}\sup_{t\in[0,\tau]}\left|\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\right|=O_{p}(1).$
(301)

We therefore have $Q_{53}=o_{p}(1)$.

For $Q_{54}$, we have $Q_{54}=Q_{541}-Q_{542}+Q_{543}$, where

$\displaystyle Q_{541}=$
$\displaystyle\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})\\}\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o}),$
$\displaystyle Q_{542}=$
$\displaystyle\int_{0}^{\tau}\\{\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})-\bar{\alpha}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o}),$
$\displaystyle Q_{543}=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\int_{0}^{\tau}\\{\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$

By (LABEL:C3third) of Assumption 11, we have $Q_{541}=o_{p}(1)$ and
$Q_{542}=o_{p}(1)$. $Q_{543}$ is again a sample average of i.i.d. terms when
conditional on $O^{\dagger}$, and each of the increments in
$D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$ is an increment of a
monotone function. So like $Q_{51}$, we apply (132), followed by the law of
total variance and have

$\displaystyle\mbox{Var}(Q_{543})\lesssim
0+\frac{n}{n}E^{\dagger}\left(E\left[\left\\{\sup_{t\in[0,\tau]}\left|\bar{\alpha}(t;\beta^{*},\hat{\pi},S^{o},\hat{S}_{c})-\bar{\alpha}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\right|\right\\}^{2}\right]\right)=o(1),$
(302)

where $o(1)$ follows from (100) of Assumption 11. Therefore,
$Q_{543}=o_{p}(1)$ by Chebyshev’s inequality and $Q_{54}=o_{p}(1)$.

Combining our results on $Q_{51}$ to $Q_{54}$ ,we have $Q_{5}=o_{p}(1)$.

Same as how we dealt with $Q_{5}$, we can decompose $Q_{6}$ in a similar way
and show that each of the terms is $o_{p}(1)$, so we omit the details here.

Lastly, we consider $\sqrt{n}U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})$. Using
(276), we have

$\displaystyle\sqrt{n}U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})$ (303)
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\left[D_{2i}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})-\int_{0}^{\tau}\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})\right]$
(304) $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi_{i}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})$
(305)
$\displaystyle+\int_{0}^{\tau}\\{\bar{\alpha}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})-\bar{A}(t;\beta^{*},\pi^{o},S^{o},S_{c}^{o})\\}\cdot\frac{1}{\sqrt{n}}\sum_{i=1}^{n}D_{1i}(t;\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o}).$
(306)

From (LABEL:C3third) of Assumption 11, we have $\eqref{AIPW:l4.3}=o_{p}(1)$,
so we have

$\displaystyle\sqrt{n}U(\beta^{*},\pi^{o},S^{o},S_{c}^{o})=\frac{1}{\sqrt{n}}\sum_{i=1}^{n}\psi_{i}(\beta^{*},\Lambda^{*},\pi^{o},S^{o},S_{c}^{o})+o_{p}(1).$
(308)

∎

###  F.4 Consistency of $\hat{\Lambda}(t)$

Given the consistency of $\hat{\beta}$, Assumptions 1-6 and C1-C3 from Cox
MSM, when either $S=S^{o}$, or $(\pi,S_{c})=(\pi^{o},S_{c}^{o})$, we show that
for each $t$,

$\displaystyle\hat{\Lambda}(t;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})\overset{p}{\to}\Lambda^{*}(t),$
(309)

where

$\displaystyle\hat{\Lambda}(t;\beta,\pi,S,S_{c})=\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\pi,S,S_{c})}{\mathcal{S}^{(0)}(u;\beta,\pi,S,S_{c})}.$
(310)

and

$\displaystyle\Lambda^{*}(t)=\int_{0}^{t}\frac{\sum_{a=0,1}dF_{a}(t)}{\sum_{a=0,1}S_{a}(t)e^{\beta^{*}a}}=-\int_{0}^{t}\frac{\sum_{a=0,1}dS_{a}(t)}{\sum_{a=0,1}S_{a}(t)e^{\beta^{*}a}}$
(311)

Proof.

Let
$\hat{\Lambda}(t;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})-\Lambda^{*}(t)=L_{1}+L_{2}+L_{3}$,
where

$\displaystyle
L_{1}=\hat{\Lambda}(t;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})-\hat{\Lambda}(t;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})$
(312) $\displaystyle
L_{2}=\hat{\Lambda}(t;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})-\hat{\Lambda}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})$
(313) $\displaystyle
L_{3}=\hat{\Lambda}(t;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\Lambda^{*}(t).$
(314)

Consider $L_{1}$, which can be written as

$\displaystyle L_{1}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\hat{\pi},\hat{S},\hat{S}_{c})}{\mathcal{S}^{(0)}(u;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})}\cdot\\{\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})-\mathcal{S}^{(0)}(u;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})\\}$
(315) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\hat{\pi},\hat{S},\hat{S}_{c})}{\mathcal{S}^{(0)}(u;\hat{\beta},\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})}$
(316)
$\displaystyle\times\frac{1}{n}\sum_{j=1}^{n}\Bigg{[}\frac{A^{l}_{j}Y_{j}(t)\\{e^{\beta^{*}A_{j}}-e^{\hat{\beta}A_{j}}\\}}{\pi(Z_{j})^{A_{j}}\\{1-\pi(Z_{j})\\}^{1-A_{j}}S_{c}(t;A_{j},Z_{j})}-\frac{A^{l}_{j}S(t;A_{j},Z_{j})\\{e^{\beta^{*}A_{j}}-e^{\hat{\beta}A_{j}}\\}}{\pi(Z_{j})^{A_{j}}\\{1-\pi(Z_{j})\\}^{1-A_{j}}}$
(317)
$\displaystyle\quad+\sum_{a=0,1}a^{l}\left\\{1+\frac{A_{j}^{a}(1-A_{j})^{1-a}}{\pi(Z_{j})^{a}\\{1-\pi(Z_{j})\\}^{1-a}}J_{j}(t;a,S,S_{c})\right\\}S(t;a,Z_{j})\\{e^{\beta^{*}a}-e^{\hat{\beta}a}\\}\Bigg{]}$
(318)

Since $e^{\beta^{*}a}-e^{\hat{\beta}a}=o_{p}(1)$ and
$e^{\beta^{*}A_{j}}-e^{\hat{\beta}A_{j}}=o_{p}(1)$, while the other terms are
bounded a.s., we can apply the inequality

$\displaystyle\left|\int_{a}^{b}f(t)dG(t)\right|\leq\sup_{t\in[a,b]}|f(t)|\cdot|G(b)-G(a)|$
(319)

and have $L_{1}=o_{p}(1)$.

Consider $L_{2}$, which can be written as

$\displaystyle L_{2}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\hat{\pi},\hat{S},\hat{S}_{c})-d\mathcal{N}^{(0)}_{i}(u;\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})}$
(320) $\displaystyle+$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}\cdot\\{\mathcal{S}^{(0)}(u;\beta^{*},\hat{\pi},\hat{S},\hat{S}_{c})-\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\\}.$
(321)

As we can see, we are dealing with differences between
$\hat{\pi},\hat{S},\hat{S}_{c}$ and $\pi^{*},S^{*},S_{c}^{*}$, which is the
same as in Proof of Lemma 7, so we omit the details for showing
$L_{2}=o_{p}(1)$ here.

Lastly, for $L_{3}$, we first note that from the definition of $D_{1}$, we
have

$\displaystyle L_{3}=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}-\Lambda^{*}(t)$
(322) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{d\mathcal{N}^{(0)}_{i}(u;\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}-\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{\Gamma_{i}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}d\Lambda^{*}(t)$
(323) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{D_{1i}(u;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}$
(324) $\displaystyle=$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{D_{1i}(u;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})}{\mathpzc{s}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}$
(325) $\displaystyle+$
$\displaystyle\frac{1}{n}\sum_{i=1}^{n}\int_{0}^{t}\frac{D_{1i}(u;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})}{\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\mathpzc{s}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})}\cdot\\{\mathpzc{s}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\\}.$
(326)

From the Theorem of DR,
$D_{1i}(u;\beta^{*},\Lambda^{*},\pi^{*},S^{*},S_{c}^{*})$ is mean-zero, while
$\mathpzc{s}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})$ is a fixed function,
so the entire integral in (325) is mean-zero, and by the weak law of large
numbers (325) is $o_{p}(1)$. (326) is $o_{p}(1)$ because
$\sup_{t\in[0,\tau]}\left|\mathpzc{s}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})-\mathcal{S}^{(0)}(u;\beta^{*},\pi^{*},S^{*},S_{c}^{*})\right|=o_{p}(1)$.
So $L_{3}=o_{p}(1)$.

∎