As A Dynamical System
The Bernoulli process can also be understood to be a dynamical system, specifically, a measure-preserving dynamical system. This arises because there is a natural translation symmetry on the (two-sided) product space given by the shift operator
The measure is translation-invariant; that is, given any cylinder set, one has
and thus the Bernoulli measure is a Haar measure.
The shift operator should be understood to be an operator acting on the sigma algebra, so that one has
In this guise, the shift operator is known as the transfer operator or the Ruelle-Frobenius-Perron operator. It is interesting to consider the eigenfunctions of this operator, and how they differ when restricted to different subspaces of . When restricted to the standard topology of the real numbers, the eigenfunctions are curiously the Bernoulli polynomials! This coincidence of naming was presumably not known to Bernoulli.
The coin flips of the Bernoulli process are presumed to be independent, and perfectly uncorrelated. It is reasonable to ask what might happen if they were correlated, but still time-invariant. In this case, one gets a specific kind of Markov chain, known as the one-dimensional Ising model.
Read more about this topic: Bernoulli Process
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