# ON GENERALIZED MAX-LINEAR MODELS IN MAX-STABLE RANDOM FIELDS 

Abstract. In practice, it is not possible to observe a whole max-stable ran- dom field. Therefore, a way how to reconstruct a max-stable random field in C ( [0 , 1] $^{k}$) by interpolating its realizations at finitely many points is proposed. The resulting interpolating process is again a max-stable random field. This approach uses a generalized max-linear model. Promising results have been established in the case k = 1 in a previous paper. However, the extension to higher dimensions is not straightforward since we lose the natural order of the index space. 

MICHAEL FALK, MAXIMILIAN ZOTT 

## 1. Introduction and Preliminaries 

Dombry et al. (2013) derive an algorithm to sample from the regular conditional distribution of a max-stable random field η , say, given the marginal observations η$_{s}$$_{1}$ = z$_{1}$, . . . , η$_{s}$$_{k}$ = z$_{k}$ for some z$_{1}$, . . . , z$_{d}$ from the state space and k locations s$_{1}$, . . . , s$_{d}$ . This, clearly, concerns the distribution of η and derived distributional parameters. 

Different to that, we try to reconstruct η from the observations η$_{s}$$_{1}$ , . . . , η$_{s}$$_{k}$ . This is done by a generalized max-linear model in such a way, that the interpolating process ˆ η is again a (standard) max-stable random field. 

As our approach is deterministic, once the observations η$_{s}$$_{1}$ = z$_{1}$, . . . , η$_{s}$$_{k}$ = z$_{k}$ are given, a proper way to measure the performance of our approach is the mean squared error (MSE). Convergence of the pointwise MSE as well as of the integrated 

