in Appendix B.) We also prove that this distributed sequential estimation procedure retains some of the conventional statistical properties of sequential fixed-sized confidence set estimation, and summarize these properties as per the following theorem. (The proof of Theorem \[Th1\] is in Appendix B.)

\[Th1\] Assume that for each $j$, $\{(x_{ji}, y_{ji}), i \geq 1\}$ follows the linear regression model (\[lm\]), and\
[*(A1)*]{} $\lim_{n\rightarrow \infty}\sum_{i=1}^{n}{\bm{{x}}_{ji}}{\bm{{x}}_{ji}^{\top}}/n=\Sigma$, where the matrix $\Sigma$ is positive-definite.\
Let $N_j$ be defined as in (\[Ndef\]). Then,

-   $\lim_{d\rightarrow 0}\frac{d^2N^*}{a^2\sigma^2\mu}=1 ~~~~\text{\rm almost surely}$, \[Fconclu1\]

-   $\lim_{d\rightarrow 0} P({\bm{{\beta}}_{0}} \in R_N^*)=1-\alpha$, \[sconclu2\]

-   $\lim_{d\rightarrow 0}\frac{d^2E(N^*)}{a^2\sigma^2\mu}=1$, \[Fconclu3\]

where $\mu$ is the minimum eigenvalue of matrix $\Sigma$.

Theorems \[Th1\] (2) and (3) state that the proposed method has the prespecified coverage probability $1-\alpha$, and that the ratio of total data used in all procedures to the ‘theoretical’ one—the best but unknown one—is asymptotically equal to 1. @Chow1965 first named these two properties [*asymptotic consistency*]{} and [*asymptotic efficiency*]{}, respectively, to describe the asymptotic properties of a sequential estimation procedure.

Under the current setup, the conventional sequential estimation is a case with $M=1$. The stopping time in this case is $$\begin{aligned}
\label{sNdef}
    N\equiv\inf\left\{n\geq n_0:\mbox{and}~~(\hat\sigma_n^2+\frac{1}{n})\leq \frac{d^2n}{a^2\mu_n}\right\}.\end{aligned}$$ Obviously, $a^2>\tilde{a}_j^2$ implies that $N$ is stochastically larger than 