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Turning a Coin over Instead of Tossing It

Engländer, János and Volkov, Stanislav LU (2016) In Journal of Theoretical Probability
Abstract

Given a sequence of numbers (Formula presented.) in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability (Formula presented.), (Formula presented.), independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as (Formula presented.)? We show that a number of phase transitions take place as the turning gets slower (i. e., (Formula presented.) is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is (Formula presented.). Among the scaling limits, we obtain uniform,... (More)

Given a sequence of numbers (Formula presented.) in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability (Formula presented.), (Formula presented.), independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as (Formula presented.)? We show that a number of phase transitions take place as the turning gets slower (i. e., (Formula presented.) is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is (Formula presented.). Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Central Limit Theorem, Coin tossing, Laws of Large Numbers
in
Journal of Theoretical Probability
pages
22 pages
publisher
Kluwer
external identifiers
  • scopus:84997206795
ISSN
0894-9840
DOI
10.1007/s10959-016-0725-1
language
English
LU publication?
yes
id
f42564ea-7c1e-44e6-b09c-39b5dee12f73
date added to LUP
2016-12-09 08:51:52
date last changed
2017-03-15 11:31:24
@article{f42564ea-7c1e-44e6-b09c-39b5dee12f73,
  abstract     = {<p>Given a sequence of numbers (Formula presented.) in [0, 1], consider the following experiment. First, we flip a fair coin and then, at step n, we turn the coin over to the other side with probability (Formula presented.), (Formula presented.), independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as (Formula presented.)? We show that a number of phase transitions take place as the turning gets slower (i. e., (Formula presented.) is getting smaller), leading first to the breakdown of the Central Limit Theorem and then to that of the Law of Large Numbers. It turns out that the critical regime is (Formula presented.). Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.</p>},
  author       = {Engländer, János and Volkov, Stanislav},
  issn         = {0894-9840},
  keyword      = {Central Limit Theorem,Coin tossing,Laws of Large Numbers},
  language     = {eng},
  month        = {11},
  pages        = {22},
  publisher    = {Kluwer},
  series       = {Journal of Theoretical Probability},
  title        = {Turning a Coin over Instead of Tossing It},
  url          = {http://dx.doi.org/10.1007/s10959-016-0725-1},
  year         = {2016},
}