DIFFERENTIABILITY-FREE CONDITIONS ON THE FREE-ENERGY FUNCTION IMPLYING LARGE DEVIATIONS

Abstract

Let (μα) be a net of Radon sub-probability measures on ℝ, and (tα) be a net in ]0, 1] converging to 0. Assuming that the generalized log-moment generating function L(λ) exists for all λ in a nonempty open interval G, we give conditions on the left or right derivatives of L|G, implying a vague (and thus narrow when 0 ∈ G large deviation principle. The rate function (which can be nonconvex) is obtained as an abstract Legendre–Fenchel transform. This allows us to strengthen the Gartner–Ellis theorem by weakening the essential smoothness assumption. A related question of R. S. Ellis is solved.