samples drawn uniformly from are representative of as a whole.) In the present example, a sequence of coin flips, where half are heads, and half are tails, is a "typical" sequence. There are several important points to be made about the Bernoulli process. If one writes 0 for tails and 1 for heads, one gets the set of all infinite strings of binary digits. These correspond to the base-two expansion of real numbers. Explicitly, given a sequence , the corresponding real number is The statement that the Bernoulli process is ergodic is equivalent to the statement that the real numbers are uniformly distributed. The set of all such strings can be written in a variety of ways: This set is the Cantor set, sometimes called the Cantor space to avoid confusion with the Cantor function In the end, these are all "the same thing". The Cantor set plays key roles in many branches of mathematics. In recreational mathematics, it underpins the period-doubling fractals; in analysis, it appears in a vast variety of theorems. A key one for stochastic processes is the Wold decomposition, which states that any stationary process can be decomposed into a pair of uncorrelated processes, one deterministic, and the other being a moving average process. The Ornstein isomorphism theorem states that every stationary stochastic process is equivalent to a Bernoulli scheme (a Bernoulli process with an N-sided (and possibly unfair) gaming die). Other results include that every non-dissipative ergodic system is equivalent to the Markov odometer, sometimes called an "adding machine" because it looks like elementary-school addition, that is, taking a base-N digit sequence, adding one, and propagating the carry bits. The proof of equivalence is very abstract; understanding the result is not: by adding one at each time step, every possible state of the odometer is visited, until it rolls over, and starts again. Likewise, ergodic systems visit each state, uniformly, moving on to the next, until they have all been visited. Systems that generate (infinite) sequences of N letters are studied by means of symbolic dynamics. Important special cases include subshifts of finite type and sofic systems. History and etymology The term ergodic is commonly thought to derive from the Greek words (ergon: "work") and (hodos: "path", "way"), as chosen by Ludwig Boltzmann while he was working on a problem in statistical mechanics. At the same time it is also claimed to be a derivation