Statement of theorem
Let be a measure-preserving transformation on the probability space , and let be a sequence of functions such that (subadditivity relation). Then
for -a.e. x, where g(x) is T-invariant. If T is ergodic, then g(x) is a constant.
In mathematics, Kingman's subadditive ergodic theorem is one of several ergodic theorems. It can be seen as a generalization of Birkhoff's ergodic theorem. [1] Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic). [2] As a result, it can be rephrased in the language of probability, e.g. using a sequence of random variables and expected values. The theorem is named after John Kingman.
Let be a measure-preserving transformation on the probability space , and let be a sequence of functions such that (subadditivity relation). Then
for -a.e. x, where g(x) is T-invariant. If T is ergodic, then g(x) is a constant.
Taking recovers Birkhoff's pointwise ergodic theorem.
Kingman's subadditive ergodic theorem can be used to prove statements about Lyapunov exponents. It also has applications to percolations and probability/random variables. [3]
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