2 1. BASICS ON LARGE DEVIATIONS exists as an extended real number. Further, the origin belongs to the interior o of the domain = R Λ(θ) ∞} of the function Λ(θ). By older inequality, Λ(θ) is a convex function. Consequently, Do Λ = (a, b) for some −∞ a 0 b and Λ(θ) is continuous in its domain DΛ. Define the Fenchel-Legendre transform Λ∗(λ) of Λ(θ) as (1.1.2) Λ∗(λ) = sup θ∈R θλ Λ(θ) λ R. To discuss the role played by the function Λ∗(·), we introduce the following definition. Definition 1.1.2. A function I: R −→ [0, ∞] is called a rate function if it is lower semi-continuous: For any l 0, the level set Il = R I(λ) l} is a closed set. Further, a rate function is said to be good if every level set is compact in R. We point out an equivalent statement of lower semi-continuity: For any λn,λ R with λn λ, (1.1.3) lim inf n→∞ I(λn) I(λ). We now claim that under Assumption 1.1.1, the function Λ∗(·) is a good rate function. Indeed, by definition Λ∗(λn) θλn Λ(θ) θ R n = 1, 2, · · · , which leads to lim inf n→∞ Λ∗(λn) θλ Λ(θ) θ R. Taking the supremum over θ R on the right-hand side proves (1.1.3). Let l 0 be fixed. By Assumption 1.1.1 and by the continuity of Λ(θ) in its domain there is δ 0 such that c sup |θ|≤δ Λ(θ) ∞. Consequently, for any λ with Λ∗(λ) l, l sup |θ|≤δ θλ Λ(θ) δ|λ| c. Therefore, the level set Λa = R, Λ∗(λ) l} is compact. In addition, the fact that Λ∗(λ) is the conjugate of the convex function Λ(θ) makes Λ∗(λ) a convex function. Definition 1.1.3. A convex function Λ: R −→ (−∞.∞] is essentially smooth if: (1) o = (a, b) is non-empty.
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