# Is there a continuous-time version of Kingman's subadditive decomposition theorem?

Kingman's subadditive ergodic theorem (see this article) states that if $x_{m,n}$ is a real valued process indexed on the set of pairs of non-negative integers $m < n$ satisfying:

1. $x_{l,n} \le x_{l,m} + x_{m,n}$ for all $0 \le l < m < n$
2. The process is stationary in the sense that the distribution of $\{x_{m,n}\}$ equals that of the shifted process $\{x_{m+1,n+1}\}$
3. $\mathbb{E}(|x_{0,1}|) <+\infty$
4. The limit $\lim_{n \to +\infty}\mathbb{E}(x_{0,n}/n)$ is finite (this is called the "time-constant" of the process).

Then the limit $\lim_{n \to +\infty}x_{0,n}/n$ exists almost surely and in mean.

The original proof was based on a decomposition theorem, which states that in fact there is a stationary additive process (i.e. satisfying $y_{m,n} = y_{m,m+1}+\cdots + y_{n-1,n}$) having the same time-constant as $\{x_{m,n}\}$ and such that $y_{m,n} \le x_{m,n}$. This reduced the proof of the subadditive ergodic theorem to Birkhoff's ergodic theorem (plus a somewhat tricky usage of Fatou's lemma and the likes).

The subadditive ergodic theorem has an extension to processes indexed on pairs of non-negative real numbers (which in fact Kingman himself proved). The proof is simply to discretize the process and apply the discrete-time version (the condition $\mathbb{E}(\sup_{0 \le s < t \le 1}|x_{s,t}|) < +\infty$ is needed instead of 3. or there are counter-examples).

My question is if anybody knows of a published continuous-time version of the decomposition theorem.

Also, does anybody know any proofs of the decomposition theorem besides that of the original paper by Kingman, the one by Burkholder, and the one by del Junco?

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Look into Jean Bertoin's lecture notes for the PIMS summer school 2010. – John Jiang Sep 23 '10 at 1:26
John Jiang: Thanks for the reference. As far as I can tell the lecture notes are about the Coalescent random tree process (another nice theory initiated by Kingman). I might be missing something but I think there's not much subadditive ergodic theory in there. – Pablo Lessa Sep 23 '10 at 19:52
If you have access to Random Dynamical Systems by Arnold, he states and proves a continuous-time version of Kingman's Subadditive Ergodic Theorem (see Theorem 3.1.6 on page 123). On page 117 there is a list of other references for Kingman's SET. In particular, he cites this book Ergodic Theorems by Krengel. I haven't looked at any of that in much detail though. – orlandoweber Apr 5 '13 at 4:40
@orlandoweber: Thanks for the reference. In my version of Arnold's book he talks only about the Subaddive Ergodic Theorem. The continuous time version of this theorem is well known and in fact included in Kingman's article in Annals of Probability. My question is about the much more obscure Subadditive Decomposition Theorem. – Pablo Lessa Apr 8 '13 at 18:26