Kingman's subadditive ergodic theorem

Let ${\displaystyle T}$ be a measure-preserving transformation on the probability space ${\displaystyle (\Omega ,\Sigma ,\mu )}$, and let ${\displaystyle \{g_{n}\}_{n\in \mathbb {N} }}$ be a sequence of ${\displaystyle L^{1}}$ functions such that ${\displaystyle g_{n+m}(x)\leq g_{n}(x)+g_{m}(T^{n}x)}$ (subadditivity relation). Then

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}(x)}{n}}=:g(x)\geq -\infty }$

for ${\displaystyle \mu }$-a.e. x, where g(x) is T-invariant. If T is ergodic, then g(x) is a constant.

Taking ${\displaystyle g_{n}(x):=\sum _{j=0}^{n-1}f(T^{j}x)}$ 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]

• Theorem proof (Steele)