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

Abstract

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 {\bf 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 $\varepsilon_i^*$, and, $(ii)$ a canonical ensemble satisfying the Shannon-Jaynes maximum entropy theory for energy-eigenvalues $\varepsilon_i$. Such an invertible mapping is demonstrated to facilitate an \emph{implicit} reconciliation between the $q^*$-maximum entropy principle and the large deviation theory. Numerical examples for exemplary cases are provided.