Kingman subadditive ergodic theorem

Statement

Let $T$ preserve a probability measure and let $(a_n)$ be a real subadditive cocycle with $a_n\in L^1$ for every $n\geq1$ . Assume

\inf_{n\geq1}\frac1n\int_Xa_n\,d\mu>-\infty.

Then $a_n/n$ converges almost everywhere and in $L^1$ to an integrable invariant function $a_\infty$ , and

\int a_\infty\,d\mu=\inf_{n\geq1}\frac1n\int a_n\,d\mu.

If $T$ is ergodic, this limit is the displayed constant almost everywhere.

Applications

Additive orbit sums recover Birkhoff's theorem. Logarithms of matrix-product norms give asymptotic growth rates and are a starting point for the multiplicative ergodic theorem. The lower-bound hypothesis here ensures a finite integrable limit; more general versions can allow a limit of $-\infty$ .

References

Anders Karlsson, “A proof of the subadditive ergodic theorem”, §1, theorem attributed to Kingman (1968).