Bernoulli Process - As The Cantor Space

As The Cantor Space

The space is equivalent to the Cantor set, and, in formal discussions, it is often called the Cantor space. Elements of the Cantor set are the infinitely long strings of H, T. The above discussion shows that the Bernoulli process is one particular kind of measure on the Cantor space, although there are many others.

The Cantor space is universal in many ways; one particular way in which this holds is that the real numbers, specifically, the unit interval can be embedded in the Cantor set. One does this by interpreting coin flips H and T as 0 and 1, and then takes an infinite sequence of these as a binary number. That is, given an infinite sequence of binary digits, one considers

This function is onto but not one-to-one; every dyadic rational has two possible representations, one ending with all zero's and one ending with all one's. As real numbers, these are the same; this is commonly known as the theorem that 0.999...=1.000....

The shift operator composed with this map gives the Bernoulli map. That is, one has

where denotes the floor of 2y.

In order to study this map properly, one should, again, consider not infinite sequences of coin-tosses, but rather, the finite sequences that lead to the product topology of the Bernoulli process. In this case, one finds that the Bernoulli map is ergodic, but not strong mixing.

The analogous construction for the two-sided Bernoulli process results in the Baker's map. Thus, the Bernoulli process is an Axiom A system.