Kingman’s subadditive ergodic theorem
Let be a probability space, and be a measure preserving dynamical system. let , be a subadditive sequence of measurable functions, such that is integrable, where . Then, the sequence converges almost everywhere to a function such that:
is invariant, that is, for almost all , and
The fact that the limit equals the infimum is a consequence of the fact that the sequence is a subadditive sequence and Fekete’s subadditive lemma.
A superadditive version of the theorem also exists. Given a superadditive sequence , then the symmetric sequence is subadditive and we may apply the original version of the theorem.
Every additive sequence is subadditive. As a consequence, one can prove the Birkhoff ergodic theorem from Kingman’s subadditive ergodic theorem.
|Title||Kingman’s subadditive ergodic theorem|
|Date of creation||2014-03-18 14:34:03|
|Last modified on||2014-03-18 14:34:03|
|Last modified by||Filipe (28191)|
|Related topic||birkhoff ergodic theorem|