# Subadditive Kingmans theorem for lattices.

I am looking for a multidimensional version of Kingman's subadditive theorem. I found this but it is not exactely what I need.

I would rather have something like that:

Let us consider $\mathbb{Z}^2_+$ and a family $(X)$ of random variables indexed by pairs of points in $\mathbb{Z}^2$ i.e. $X_{z_1, z_2}$ is a random variable associated with subgrid of $\mathbb{Z}^2$ "starting from" $z_1$ and ending on $z_2$. Assume that for any rectangular subgrid $\Lambda \subset \mathbb{Z}^2_+$ and point $x\in \Lambda$ we have

$X_{\Lambda}\leq X_{\Lambda_1} + X_{\Lambda_2}+X_{\Lambda_3}+X_{\Lambda_4},$

where $\Lambda_1,\Lambda_2,\Lambda_3,\Lambda_4$ is $\Lambda$ split in point $x$ into four subgrids. I suppose that this together with some "usual" ergodic theorem conditions should imply that

$X_{(0,0),(n,n)}/n^2$ converges in $L^1$ and a.s.

May be some one you could give me some references.

Being in the topic of subadditivity. I found a multidimensional version of Fekete lemma. It is surprising for me that it was not done before 2007. But may be it was. Again I will be happy to know any.

## 1 Answer

Have a look at the paper by Nguyen, Xuan-Xanh Ergodic theorems for subadditive spatial processes, Z. Wahrsch. Verw. Gebiete 48 (1979), no. 2, 159–176, MR0534842 (82c:60056). The setup there is somewhat different, but it seems that the main result there should contain your claim.