The fair coin toss is an example of a Bernoulli shift and is an ergodic process. After any given number n tosses of being either head or tails the probability approaches 1/2 in the limit. By Poincaré recurrence theorem there will be recurrence relation for measure-preserving dynamic systems. This very small oscillating error describes both the approaching of 1/2 by being redshifted and a recurrence relation by passing through the x-axis. See “Modern ergodic theory, from physics hypothesis to mathematical theory” slides by Doğan Çömez. As an aside also consider the connection mentioned there near the end between stochastic processes, the Cantor set, and fractals.
I indeed first wanted to describe the oscillation phenomena using the sum of the Cantor sequence (Ca(n) is zero or one depending whether the ternary expansion of n contains a 1 or not). This sum grows having an “expected” factor n^(log_3(2)) with ‘log_3(2)=fractal dim of the Cantor set’ and an oscillation term which is insane: its a devil’s staircase type function. But I decided tossing coins is easier 🤣
.... wow that's weird. cool video! I kind of want to simulate this and see it happen, but if it's a 1x10^-6 error it would take something on the order of a million squared samples to see the behaviour, right? that would take an awful long time ...
The fair coin toss is an example of a Bernoulli shift and is an ergodic process. After any given number n tosses of being either head or tails the probability approaches 1/2 in the limit. By Poincaré recurrence theorem there will be recurrence relation for measure-preserving dynamic systems. This very small oscillating error describes both the approaching of 1/2 by being redshifted and a recurrence relation by passing through the x-axis. See “Modern ergodic theory, from physics hypothesis to mathematical theory” slides by Doğan Çömez.
As an aside also consider the connection mentioned there near the end between stochastic processes, the Cantor set, and fractals.
I indeed first wanted to describe the oscillation phenomena using the sum of the Cantor sequence (Ca(n) is zero or one depending whether the ternary expansion of n contains a 1 or not). This sum grows having an “expected” factor n^(log_3(2)) with ‘log_3(2)=fractal dim of the Cantor set’ and an oscillation term which is insane: its a devil’s staircase type function.
But I decided tossing coins is easier 🤣
@@VisualMath Tossing a coin is easier 👍
@@Jaylooker For sure 😂
.... wow that's weird. cool video! I kind of want to simulate this and see it happen, but if it's a 1x10^-6 error it would take something on the order of a million squared samples to see the behaviour, right? that would take an awful long time ...
Exactly, that is why I didn't dare to illustrate it...😨
Anyway, I am glad that you liked the video 😀