Turning a Coin over Instead of Tossing It
(2018) In Journal of Theoretical Probability 31(2). p.10971118 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.
(Less)
 author
 Engländer, János and Volkov, Stanislav ^{LU}
 organization
 publishing date
 201806
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 Central Limit Theorem, Coin tossing, Laws of Large Numbers
 in
 Journal of Theoretical Probability
 volume
 31
 issue
 2
 pages
 1097  1118
 publisher
 Springer
 external identifiers

 scopus:84997206795
 ISSN
 08949840
 DOI
 10.1007/s1095901607251
 language
 English
 LU publication?
 yes
 id
 f42564ea7c1e44e6b09c39b5dee12f73
 date added to LUP
 20161209 08:51:52
 date last changed
 20211006 04:42:59
@article{f42564ea7c1e44e6b09c39b5dee12f73, 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 = {08949840}, language = {eng}, number = {2}, pages = {10971118}, publisher = {Springer}, series = {Journal of Theoretical Probability}, title = {Turning a Coin over Instead of Tossing It}, url = {http://dx.doi.org/10.1007/s1095901607251}, doi = {10.1007/s1095901607251}, volume = {31}, year = {2018}, }