Editing Bernoulli process (section)

==Bernoulli sequence==
The term ''Bernoulli sequence'' is often used informally to refer to a [[realization (probability)|realization]] of a Bernoulli process.
However, the term has an entirely different formal definition as given below.

Suppose a Bernoulli process formally defined as a single random variable (see preceding section). For every infinite sequence ''x'' of coin flips, there is a [[sequence]] of integers

:<math>\mathbb{Z}^x = \{n\in \mathbb{Z} : X_n(x) = 1 \} \, </math>

called the ''Bernoulli sequence''{{Verify source|date=March 2010}} associated with the Bernoulli process. For example, if ''x'' represents a sequence of coin flips, then the associated Bernoulli sequence is the list of natural numbers or time-points for which the coin toss outcome is ''heads''.

So defined, a Bernoulli sequence <math>\mathbb{Z}^x</math> is also a random subset of the index set, the natural numbers <math>\mathbb{N}</math>.

[[Almost all]] Bernoulli sequences <math>\mathbb{Z}^x</math> are [[ergodic sequence]]s.{{Verify source|date=March 2010}}