A PHASE TRANSITION FOR REPEATED AVERAGES

Sourav Chatterjee; Persi Diaconis; Allan Sly; Lingfu Zhang

doi:10.1214/21-AOP1526

A PHASE TRANSITION FOR REPEATED AVERAGES

Sourav Chatterjee, Persi Diaconis, Allan Sly, Lingfu Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

Let x₁,..., x_n be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace x_I, x_J by (x_I +x_J )/2, (x_I + x_J )/2. Clearly, all the coordinates converge to (x₁ +· · ·+x_n)/n. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain

Original language	English (US)
Journal	Annals of Probability
Volume	50
Issue number	1
DOIs	https://doi.org/10.1214/21-AOP1526
State	Published - Jan 2022

All Science Journal Classification (ASJC) codes

Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Convergence rate
Cutoff phenomenon
Markov chain

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10.1214/21-AOP1526

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title = "A PHASE TRANSITION FOR REPEATED AVERAGES",

abstract = "Let x1,..., xn be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace xI, xJ by (xI +xJ )/2, (xI + xJ )/2. Clearly, all the coordinates converge to (x1 +· · ·+xn)/n. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain",

keywords = "Convergence rate, Cutoff phenomenon, Markov chain",

author = "Sourav Chatterjee and Persi Diaconis and Allan Sly and Lingfu Zhang",

note = "Funding Information: Funding. Sourav Chatterjee{\textquoteright}s research was partially supported by NSF Grant DMS-1855484. Persi Diaconis{\textquoteright}s research was partially supported by NSF Grant DMS-0804324. Allan Sly{\textquoteright}s research was partially supported by NSF Grant DMS-1855527, Simons Investigator Grant and a MacArthur Fellowship. Publisher Copyright: {\textcopyright} Institute of Mathematical Statistics, 2022",

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AU - Chatterjee, Sourav

AU - Diaconis, Persi

AU - Sly, Allan

AU - Zhang, Lingfu

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N2 - Let x1,..., xn be a fixed sequence of real numbers. At each stage, pick two indices I and J uniformly at random, and replace xI, xJ by (xI +xJ )/2, (xI + xJ )/2. Clearly, all the coordinates converge to (x1 +· · ·+xn)/n. We determine the rate of convergence, establishing a sharp “cutoff” transition answering a question of Jean Bourgain

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KW - Markov chain

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A PHASE TRANSITION FOR REPEATED AVERAGES

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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