  Maxima of moving maxima of continuous functions (CM3) are max-stable
processes aimed at modeling extremes of continuous phenomena over time. They
are defined as Smith and Weissman's M4 processes with continuous functions
rather than vectors. After standardization of the margins of the observed
process into unit-Fr\'echet, CM3 processes can model the remaining
spatio-temporal dependence structure. CM3 processes have the property of joint
regular variation. The spectral processes from this class admit particularly
simple expressions. Furthermore, depending on the speed with which the
parameter functions tend toward zero, CM3 processes fulfill the finite-cluster
condition and the strong mixing condition. For instance, these three properties
put together have implications for the expression of the extremal index.
  A method for fitting a CM3 to data is investigated. The first step is to
estimate the length of the temporal dependence. Then, by selecting a suitable
number of blocks of extremes of this length, clustering algorithms are used to
estimate the total number of different profiles. The number of parameter
functions to retrieve is equal to the product of these two numbers. They are
estimated thanks to the output of the partitioning algorithms in the previous
step. The full procedure only requires one parameter which is the range of
variation allowed among the different profiles. The dissimilarity between the
original CM3 and the estimated version is evaluated by means of the Hausdorff
distance between the graphs of the parameter functions.
