  We introduce a novel concept of convergence for Markovian processes within
Orlicz spaces, extending beyond the conventional approach associated with $L_p$
spaces. After showing that Markovian operators are contractive in Orlicz
spaces, our key technical contribution is an upper bound on their contraction
coefficient, which admits a closed-form expression. The bound is tight in some
settings, and it recovers well-known results, such as the connection between
contraction and ergodicity, ultra-mixing and Doeblin's minorisation.
Specialising our approach to $L_p$ spaces leads to a significant improvement
upon classical Riesz-Thorin's interpolation methods. Furthermore, by exploiting
the flexibility offered by Orlicz spaces, we can tackle settings where the
stationary distribution is heavy-tailed, a severely under-studied setup. As an
application of the framework put forward in the paper, we introduce tighter
bounds on the mixing time of Markovian processes, better exponential
concentration bounds for MCMC methods, and better lower bounds on the burn-in
period. To conclude, we show how our results can be used to prove the
concentration of measure phenomenon for a sequence of Markovian random
variables.
