Kingman's subadditive ergodic theorem
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. Statement of theorem. Let formula_1 be a measure-preserving transformation on the probability space formula_2, and let formula_3 be a sequence  of formula_4 functions such that formula_5 (subadditivity relation). Then for formula_7-a.e. "x", where "g"("x") is "T"-invariant. If "T" is ergodic, then "g"("x") is a constant. Applications. Taking formula_8 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.