\ldots, \mathbf{X}_{K}$, where $\mathbf{X}_{k} = (X_{n_{(k-1)}+1},
\ldots,X_{n_{(k)}})$; $n_{(k)}=\sum_{i=1}^kn_i$; $n_i$ is a total number of observations in a stage $i$, $1\le i \le  k\le K$, and $n_{(0)}=0$. Each stage terminates with an analysis that results in a decision to stop the study or to enroll a new group of patients. Every GSD stage is assumed to be “reachable”, that is, there is a positive probability of reaching each stage. Further, to simplify the material, the term *density* is used to refer to probability measures without formally distinguishing between p.m.f.s and p.d.f.s.

The assumption of a joint canonical distribution states that $\mathbf{Z}$ follows a multivariate normal distribution with a vector of means $\left(\theta\sqrt{{\cal{I}}_1},\ldots,\theta\sqrt{{\cal{I}}_K}\right)$ and a covariance matrix with elements $\sqrt{{\cal{I}}_{k_1}/{\cal{I}}_{k_2}}$, $1 \le k_1\le k_2 \le K$ \[Section 3.1 in @jennison1999\]. Normal approximations are useful for many statistical tests. For example, if a researcher is interested in estimating $\theta$ with i.i.d. observations and the MLE of $\theta$ at the end of stage $k$ is $\widehat\theta_{k}$, then under common regularity conditions, $\sqrt{n_k}(\widehat\theta_{k}-\theta)/\sigma_{\theta}\to N(0,1)$ and the statistic $Z_k = \sqrt{n_k}\ \widehat\theta_{k}/\sigma_{\theta}\to N(0,1)$ when $\theta = 0$. Then if the sample size is large enough, a normal approximation is reasonable. But when the $Z_k$ is combined 