study proceeds to the second stage.

Example: The Two-Stage One-Sided Pocock Design
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If $n_1=n_2$, then $r_{(D)} = 1 + I(D=2)$ and $V_{(D)} = I(D=1)Z_1 + I(D=2)\left(Z_1 + Z_2\right)/\sqrt{2}$. Then, $$\text{Pr}_{\theta}\left(V_{(D)} < v \right) \to (\alpha/2) \Phi\left(\left. v\right\vert Z_1>c_1 \right)
+ (1-\alpha/2) \int_{-\infty}^{c_1} \Phi\left(
\sqrt{2} v - y\right) \frac{\phi\left(y\right)}{\text{Pr}\left(y \le c_1\right)}dy.$$ Note if $Z_1\le c_1$, then $$\text{Pr}_{\theta}\left(\frac{Z_1+Z_2}{\sqrt{2}} < v \vert Z_1\le c_1\right) = 
\int_{-\infty}^{c_1} \Phi\left(
\sqrt{2} v - y\right) \frac{\phi\left(y\right)}{\text{Pr}\left(y \le c_1\right)} dy,$$ which is a continuous mixture of distributions.

Asymptotic Results under Local Alternatives for $K$-Group Sequential Designs 
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With large samples, the MLE given $D=k$ can be approximated recursively by $$\begin{aligned}
\tilde\theta_{(k)} &\approx& \frac{n_{(k-1)}}{n_{(k)}}\tilde\theta_{(k-1)} + 
                 \frac{n_k}{n_{(k)}}\widehat\theta_k \label{MLE_k}
                                \approx \sum_{j=1}^k \frac{n_j}{n_{(k)}} \widehat\theta_j,\end{aligned}$$ where $\tilde\theta_{(k-1)}$ is an MLE based on cumulative data from stages $1$ to $k-1$, and $\widehat\theta_{k}$ is the MLE based on stage $k$ data only. In $K$-stage GSDs, the limiting random variable $r_{(D)}$ generalizes to $$\label{tau_n}
\sum_{k=1}^K I(D=k)\frac{n_{(k)}}{n_{k}} \overset{d}{\to} \sum_{k=1}^K I(D=k)r_{(k)} := r_{(D)},$$ where $r_{(k)}$ is the asymptotic ratio of a cumulative and a stage-specific sample size. Equation (\[tau\_n\]) generalizes a $2$-dimensional definition of $r_{(D)}$. Thus, $r_{(D)}$ is a multinomial random variable with support on $r_{(k)}$, $k=1,\ldots,K$.

Let $n_{(k)}/n_j \to r_{(k)j} \in (0,\infty)$, $j\le k$ with $r_{(k)k} 