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A Phase Transition For Repeated K-Averages

Rohit Chaudhuri

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 $\frac{n}{k-1}\log n$.

A Phase Transition For Repeated K-Averages

Abstract

Let be a fixed sequence of real numbers. At each stage, pick integers uniformly at random without replacement and then for each replace by . It is easy to observe that all the co-ordinates converge to . In this article, we extend the result of \cite{chatterjee2019note} by establishing order of decay of the expected distance. Furthermore, we establish the mixing time to be in between and .
Paper Structure (3 sections, 3 theorems, 3 equations)

This paper contains 3 sections, 3 theorems, 3 equations.

Table of Contents

  1. Main Result
  2. PROOFS
  3. Decay of expected $L^{2}$ distance

Key Result

Theorem 1.1

For $x_{0}=(1,0,\dots,0),$ in probability as $n \to \infty$ we have for any $\theta < \frac{1}{k \log k}$, and for any $\theta > \frac{1}{k-1}$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Corollary 2.2
  • proof