“…Among extensions, hidden semi-Markov chains (HSMCs) can be very useful as they allow the modelling of any sojourn time in a given class, when it is necessarily geometric in HMCs [13], [14], [15], [16]. Hidden bivariate Markov chains [17], [18], double Markov chains [19], or still pairwise Markov chains (PMCs) [20], [21], [22], [23] are other extensions. This paper is related to "triplet Markov chains" (TMCs), which is an extension of PMCs consisting in considering a third stochastic sequence, which might or might not have a practical signification, along the hidden sequence to be estimated and the sequence of observations, and assuming the joint Markovianity of the three sequences.…”