Fermions from classical probability and statistics defined by stochastic independence

The case study of fermions and the attempt to deduce their structure from classical probability opens new ways for classical and quantum probability, in particular, for the notion of stochastic coupling which, on the basis of the example of fermions, we enlarge to the notion of algebraic coupling, and for the various notions of stochastic independence. These notions are shown to be strictly correlated with algebraic and stochastic couplings. This approach allows to expand considerably the notion of open system. The above statements will be illustrated with some examples. The last section shows how, from these new stochastic couplings, new statistics emerge alongside the known Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac statistics.