a martingale difference sequence with respect to an increasing sequence of $\sigma$-fields $\{{\cal{F}}_n=\sigma\{(x_j, y_j): j=1, \ldots, n\}\}$ with $$\begin{aligned}
    \sup_nE(|\epsilon_n|^\alpha |{\cal{F}}_{n-1})<\infty \mbox{ almost surely for some
    }\alpha>2,\nonumber\end{aligned}$$

-   [(A3)]{} The maximum and minimum eigenvalues of design matrix $\sum_{i=1}^{n}x_ix_i^{T}$ satisfy, with a probability of 1, that $$\begin{aligned}
    \lambda_{min}(\sum_{i=1}^{n}x_ix_i^{T})\rightarrow \infty \text{ and }
    \log(\lambda_{max}(\sum_{i=1}^{n}x_ix_i^{T}))=o(\lambda_{min}(\sum_{i=1}^{n}x_ix_i^{T})).\nonumber $$

These conditions are very general, and in their paper, they do not refer to any particular design. Following their results, we will show later that if we adopt the D-optimality of statistical experimental design methods to locate new observations from the data pool, these conditions will be satisfied. Hence, with the previously defined stopping criterion, the parameter estimates will have asymptotic consistency and efficiency, as stated below.

\[Th2\] Assume that conditions in Theorem 1, (A2), and (A3) hold. Let $N_j$ be defined as in (\[Ndef\]), where new samples are selected by the aforementioned D-optimality. Then, we have $\lim_{d\rightarrow 0}(d^2N^*)/(a^2\sigma^2\mu)=1$ almost surely, $\lim_{d\rightarrow 0} P(\beta_0 \in R_N^*)=1-\alpha$ and $\lim_{d\rightarrow 0}[d^2E(N^*)]/(a^2\sigma^2\mu)=1$, where [$\mu$ is the minimum eigenvalue]{} of matrix $\Sigma$.

Theorem \[Th2\] simply says that with adaptive designs that use the $D$-optimality criterion, we still have asymptotic properties similar to those in Theorem 1. (The proof of this theorem 