depends on parameters \[e.g., @Wang14\].

All these specific sets of assumptions include assumptions of the existence and consistency of the MLE.

Asymptotic Results under Local Alternatives for Two-Stage Group Sequential Designs
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With large sample sizes, due to , MLEs calculated on $n_1+n_2$ independent and identically distributed observations can be approximated using $$\tilde\theta_{(2)} = \frac{n_1}{n_1+n_2}\widehat\theta_1 + \frac{n_2}{n_1+n_2}\widehat\theta_2 + o_p\left(n_1^{-0.5}\right)$$ if data collection did not stop at stage $1$, otherwise use $\tilde\theta_{(1)} = \widehat\theta_1$. Then, with a random stopping index $D$, $$\tilde\theta_{(D)} = I(D=1)\tilde\theta_{(1)} + I(D=2)\tilde\theta_{(2)}.$$

As shown in Theorem 1 in @Tarima2019, the asymptotic properties of the standardized estimate $$\begin{aligned}
V_{(D)} &:= \sqrt{n_{(D}}\left(\tilde\theta_{(D)} - \theta\right)/\sigma_{\theta} \notag \\
&= I(D=1)\sqrt{n_1}\left(\tilde\theta_{(1)} - \theta\right)/\sigma_{\theta} + 
I(D=2)\sqrt{n_1+n_2}\left(\tilde\theta_{(2)} - \theta\right)/\sigma_{\theta}\end{aligned}$$ depend on the existence and distribution of the limiting random variable $r_{(D)}$ that is defined by $$I(D=1) + I(D=2)\frac{n_1+n_2}{n_2} \overset{d}{\to} r_{(D)},$$ as $n_1\to\infty$. Then, for a sequence of local alternative hypotheses $\theta = h/\sqrt{n_1}$, $$\begin{aligned}
\text{Pr}_{\theta}\left(V_{(D)} < v \right) &\to& p_1 \Phi\left(\left. v\right\vert D=1\right) \notag \\
&+& (1-p_1) \int_{-\infty}^{c_1} \Phi\left(
\sqrt{r_{(D)}} v - \sqrt{r_{(D)}-1} y\right) \phi\left(y|D=2\right) dy,\end{aligned}$$ where $p_1 = \lim_{n_1\to\infty}{\text{Pr}_{\theta}}\left(D=1\right)$ is the limiting stage 1 stopping probability. Using previous terminology, under $\theta = 0$, $V_{(2)}=Z_{1}=Z_{(1)}$ if stopped at stage 1, and $V_{(2)}=Z_{(2)}$, if the 