$C1$-$C3$ are standard for the consistency and asymptotic normality of maximum likelihood estimators. $C4$ is the regularity condition that guarantees the trajectories $g\left(.\mid a\right)$ depends on $a$ smoothly. $C5$ is the key to guarantee the identification of the model and , although it turns out that $C5$ can be discarded without any impact on the consistency of $\hat{b}$ and $\hat{\lambda}$, and both of $C5$ and $C6$ can be discarded when the event time and the longitudinal process satisfy a kind of Markovian property and the extra longitudinal observations are available. We will go back to these extensions in the section \[extension\].

\[theorem-identification\] Under Condition $C5$ and $C6$, model and are identifiable. And $E_{\Omega_{0}}\left(\log\left(V_{\Omega}\right)\right)$ has the unique maximal point, $\Omega_{0}$. In addition, if $C4$ holds, $E_{\Omega_{0}}\left(\log\left(V_{\Omega}\right)\right)$ is continuous with respect to the variable $\Omega$.

\[theorem-consistency\] (1). Under $C1$-$C6$, the estimator $\hat{a}$, $\hat{b}$ are consistent and $\hat{b}-b_{0}\rightarrow_{d}N\left(0,I^{-1}\right)$;

(2). the estimator $\hat{\lambda}^{s}$ converges to $\lambda_{0}$ according to the weak-$*$ topology and $\sqrt{n}\left(\int_{0}^{t}\hat{\lambda}^{s}\left(\tau\right)-\lambda_{0}\left(\tau\right)d\tau\right)$ converges weakly to a Gaussian Process;

\[theorem-ALASSO\] Under $C7$, the estimator $\hat{a}_{AL}$ and $\hat{\lambda}_{AL}^{s}$ has the same properties as $\hat{a}$ and $\hat{\lambda}^{s}$ as stated in Theorem 2, and the estimator $\hat{b}_{AL}$ has the following oracle property:

(1). denote $\mathcal{A}\subset\left\{ 1,\dots,p\right\} $ 