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.
Gerard Ben Arous. Dan Virgil Voiculescu. "Free extreme values." Ann. Probab. 34 (5) 2037 - 2059, September 2006. https://doi.org/10.1214/009117906000000016