Turning a Coin over Instead of Tossing It
(2018) In Journal of Theoretical Probability 31(2). p.1097-1118- 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
- 2018-06
- 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
- 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
- 2022-03-08 23:04:03
@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}}, keywords = {{Central Limit Theorem; Coin tossing; Laws of Large Numbers}}, language = {{eng}}, number = {{2}}, pages = {{1097--1118}}, publisher = {{Springer}}, 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}}, doi = {{10.1007/s10959-016-0725-1}}, volume = {{31}}, year = {{2018}}, }