Ergodic decomposition

Computability of the ergodic decomposition

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-...

Randomness and the Ergodic Decomposition

We briefly present ongoing work about Martin-Löf randomness and the ergodic decomposition theorem. In [ML66], Martin-Löf defined the notion of an algorithmically random infinite binary sequence. The idea was to have an individual notion of a random object, an object that is acceptable as an outcome of a probabilistic process (typically, the tossing of a coin). This notion has proved successful,...

A Guide to the Ergodic Decomposition Theorem: Ergodic Measures as Extreme Points

The ergodic decomposition of asymptotically mean stationary random sources

It is demonstrated how to represent asymptotically mean stationary (AMS) random sources with values in standard spaces as mixtures of ergodic AMS sources. This an extension of the well known decomposition of stationary sources which has facilitated the generalization of prominent source coding theorems to arbitrary, not necessarily ergodic, stationary sources. Asymptotic mean stationarity gener...

Ergodic decomposition of excess entropy and conditional mutual information∗

The article discusses excess entropy defined as mutual information between the past and future of a stationary process. The central result is an ergodic decomposition: Excess entropy is the sum of self-information of shift-invariant σ-field and the average of excess entropies for the ergodic components of the process. The result is derived using generalized conditional mutual information for fi...