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Kingman's subadditive ergodic theorem

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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. Intuitively, the subadditive ergodic theorem is a kind of random variable version of Fekete's lemma (hence the name ergodic). 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.

Contents

Statement of theorem

Let T be a measure-preserving transformation on the probability space ( Ω , Σ , μ ) , and let { g n } n ∈ N be a sequence of L 1 functions such that g n + m ( x ) ≤ g n ( x ) + g m ( T n x ) (subadditivity relation). Then

lim n → ∞ g n ( x ) n = g ( x ) ≥ − ∞

for μ -a.e. x, where g(x) is T-invariant. If T is ergodic, then g(x) is a constant.

Applications

If we take g n ( x ) := ∑ j = 0 n − 1 f ( T j x ) , then we have additivity and we get 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.

References

Kingman's subadditive ergodic theorem Wikipedia
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