Invertible mappings and the large deviation theory for the $q$-maximum entropy principle

The possibility of reconciliation between canonical probability distributions obtained from the q-maximum entropy principle with predictions from the law of large numbers when empirical samples are held to the same constraints, is investigated into. Canonical probability distributions are constrained by both: (i) the additive duality of generalized statistics and (ii) normal averages expectations. Necessary conditions to establish such a reconciliation are derived by appealing to a result concerning large deviation properties of conditional measures. The (dual) q-maximum entropy principle is shown not to adhere to the large deviation theory. However, the necessary conditions are proven to constitute an invertible mapping between: (i) a canonical ensemble satisfying the q-maximum entropy principle for energy-eigenvalues ε∗i , and, (ii) a canonical ensemble satisfying the Shannon-Jaynes maximum entropy theory for energy-eigenvalues εi. Such an invertible mapping is demonstrated to facilitate an implicit reconciliation between the q-maximum entropy principle and the large deviation theory. Numerical examples for exemplary cases are provided.