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Journal of Theoretical Probability

pp 1–22

First Online: 25 November 2016Received: 13 June 2016Revised: 07 November 2016


Given a sequence of numbers \p n {n\ge 2}\ 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 \p n\, \ge 2\, independently of the sequence of the previous terms. What can we say about the distribution of the empirical frequency of heads as ightarrow \infty \? We show that a number of phase transitions take place as the turning gets slower i. e., \p n\ 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 \p n=\text {const}-n\. Among the scaling limits, we obtain uniform, Gaussian, semicircle, and arcsine laws.

KeywordsCoin tossing Central Limit Theorem Laws of Large Numbers The hospitality of Microsoft Research and the University of Washington is gratefully acknowledged by the first author.

Research of the second author was supported in part by the Swedish Research Council Grant VR2014–5157.

Mathematics Subject Classification 201060F05 60J10  Download fulltext PDF

Autor: János Engländer - Stanislav Volkov

Fuente: https://link.springer.com/

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