Distributions derived from the maximization of Rényi-Tsallis entropy are often called Tsallis’ distributions. We first indicate that these distributions can arise as mixtures, and can be interpreted as the solution of a standard maximum entropy problem with fluctuating constraints. Considering that Tsallis’ distributions appear for systems with displaced or fluctuating equilibriums, we show that they can be derived in a standard maximum entropy setting, taking into account a constraint that displace the standard equilibrium and introduce a balance between several distributions. In this setting, the Rényi entropy arises as the underlying entropy. Another interest of Tsallis distributions, in many physical systems, is that they can exhibit heavy-tails and model power-law phenomena. We note that Tsallis’ distributions are similar to Generalized Pareto distributions, which are widely used for modeling the tail of distributions, and appear as the limit distribution of excesses over a threshold. This suggests that they can arise in many contexts if the system at hand or the measurement device introduces some threshold. We draw a possible asymptotic connection with the solution of maximum entropy. This view gives a possible interpretation for the ubiquity of Tsallis’ (GPD) distributions in applications and an argument in support to the use of Rényi-Tsallis entropies.