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?



Mean of a vector
This operation decreases the variance of the set of numbers. If you include the minimum and the maximum, the operation decreases the variance by a factor bounded by some $c(k) \lt 1$ (we can take $c(k) = 1-1/(2k)$ though that is not sharp) so the vector converges to a constant if you repeatedly include the minimum and the maximum.


You can check that many predetermined sequences of indices, such as $(1,2,3),(2,3,4),...$ will cause the vector to converge to a constant vector. The operation is linear, so it suffices to check how it acts on standard basis vectors. 


