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Tail algebras for monotone and $q$-deformed exchangeable stochastic processes

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti&#39;s theorem for this type of processes. In addition, since the vacuum state on the $q$-deformed $C^*$-algebra is the only exchangeable state when $|q|<1$, we draw our attention to its tail algebra, which turns out to obey a zero-one law.