Abstract
Free probability analogs of the basics of extreme-value theory are obtained, based on Ando's spectral order. This includes classification of freely max-stable laws and their domains of attraction, using ``free extremal convolutions'' on the distributions. These laws coincide with the limit laws in the classical peaks-over-threshold approach. A free extremal projection-valued process over a measure-space is constructed, which is related to the free Poisson point process.
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