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.

Statement of theorem

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

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

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

Applications

If we take $g_{n}(x):=\sum _{j=0}^{n-1}f(T^{n}x)$, then we have additivity and we get Birkhoff's pointwise ergodic theorem.

External links

Theorem proof (Steele)