}\! I_{jj}^*\!=\!0, 1\leq j\leq p\right\},\end{aligned}$$ where $S_{\tilde {N}}=({\bm{{O}}}{\bm{{Z}}}-{\bm{{O}}}{\tilde{{\bm{{\beta}}}}})^{\top}{\tilde{{\bm{{\Sigma}}}}}
({\bm{{O}}}{\bm{{Z}}}-{\bm{{O}}}{\tilde{{\bm{{\beta}}}}})$, $\nu_{\tilde {N}}=\lambda_{max}\left(\tilde {N}{\bm{{I}}^{*}}\left\{\sum_{j=1}^M {\bm{{X}}_{jN_j}}{\bm{{X}}_{jN_j}^{\top}}\right\}^{-1}{\bm{{I}}^{*}}\right)$ and $$\begin{aligned}
{\tilde{{\bm{{\Sigma}}}}}=\left(
  \begin{array}{cc}
    {\tilde{{\bm{{\Sigma}}}}_{11}}& 0 \\
    0 & 0 \\
  \end{array}
\right).\nonumber\end{aligned}$$ Using similar arguments, it is easy to see that the maximum axis of the confidence set $R_{\tilde {N}}$ is less than $2d$, and that the distributed ASE-based sequential procedure has the following properties. (The proofs for these properties are given in Appendix D.)

\[Th3\] Let $\tilde{N}_j$ be the stopping time defined in (\[Ndef-var\]). Assume that the conditions of Theorem \[Th1\] are satisfied; then, (1) $\lim_{d\rightarrow 0} P({\bm{{\beta}}_{0}} \in R_{\tilde{N}})=1-\alpha$, (2) $\lim_{d\rightarrow 0}{(d^2\tilde{N})}/{(a^2\sigma^2\tilde{\mu})}=1$ , and (3) $\lim_{d\rightarrow 0}{\{(d^2E(\tilde{N})\}}/{(a^2\sigma^2\tilde{\mu})}=1$, where $a^2$ is the $1-\alpha$ quantile of $\chi_{p_0}^2$ and $\tilde{\mu}$ is the maximum eigenvalue of matrix ${{\bm{{I}}_{0}}}\Sigma^{-1} {{\bm{{I}}_{0}}}$. Moreover, $\lim_{d\rightarrow 0}\hat p_0=p_0$ [almost surely]{} and $\lim_{d\rightarrow 0}E(\hat p_0)=p_0$.

The divide-and-conquer method is a general concept and it can usually be applied to most procedures. However, there is no fixed sample size procedure that can concurrently guarantee the accuracy (i.e. size of confidence set) and the coverage probability of the estimate. Hence, we cannot apply the same idea to any fixed sample size procedure.

Numerical studies
=================

We apply our methods to an appliance energy consumption data set 