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

TL;DR

This paper computes tail algebras for exchangeable monotone and $q$-deformed processes, establishing an analogue of de Finetti's theorem and revealing a zero-one law for the tail algebra in the $q$-deformed case.

Contribution

It introduces the computation of tail algebras for these processes and proves a de Finetti-type theorem, highlighting the zero-one law for the $q$-deformed vacuum state.

Findings

Tail algebras are computed for monotone processes.

An analogue of de Finetti's theorem is established.

The tail algebra obeys a zero-one law for $|q|<1$.

Abstract

We compute the tail algebras of exchangeable monotone stochastic processes. This allows us to prove the analogue of de Finetti'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.