Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$

I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$.

If I do it iteratively step by step, in each step we pick three components, $a_i,a_j$ and $a_k$ that are not all equal, and we replace them by their mean, $s_1=\frac{a_i+a_j+a_k}{3}$, to obtain $\mu_1 = (a_1, \ldots, a_{i-1}, s_1, a_{i+1}, \ldots, a_{j-1}, s_1, a_{j+1}, \ldots, a_{k-1}, s_1, a_{k+1}, \ldots, a_n)$.

In the next step, we select three other compounents (always not all equal) and compute $\mu_2$

By iterating, Can we have $\mu_n \rightarrow \mu$? if yes, how to pick up the three elements in each step?

Does this *"partial averaging"* have a particular name/theorem in number theory?