A Phase Transition For Repeated K-Averages

TL;DR

This paper analyzes a repeated averaging process on a fixed sequence, establishing the decay rate of the expected squared distance and the mixing time bounds, extending previous results.

Contribution

It extends prior work by determining the decay rate of the expected L^2 distance and providing bounds on the mixing time for the process.

Findings

Expected L^2 distance decay rate established.

Mixing time bounds between (n/(k log k)) log n and (n/(k-1)) log n.

Process converges to the average of initial sequence.

Abstract

Let $x_1,\dots,x_{n}$ be a fixed sequence of real numbers. At each stage, pick $k$ integers $\{I_{i}\}_{1\leq i \leq k}$ uniformly at random without replacement and then for each $i \in \{1,2,\dots,k\}$ replace $x_{I_i}$ by $(x_{I_1}+x_{I_2}+\dots+x_{I_k})/k$. It is easy to observe that all the co-ordinates converge to $(x_1+\dots+x_n)/n$. In this article, we extend the result of \cite{chatterjee2019note} by establishing order of decay of the expected $L^{2}$ distance. Furthermore, we establish the mixing time to be in between $\frac{n}{k \log k}\log n$ and $\frac{n}{k-1}\log n$.