1.1. GARTNER-ELLIS ¨ THEOREM 3 (2) Λ(θ) is differentiable in DΛ.o (3) Λ(·) is steep: lim θ→a+ Λ (θ) = lim θ→b− Λ (θ) = ∞. Theorem 1.1.4. Let Assumption 1.1.1 hold. For any closed set F ⊂ R, (1.1.4) lim sup n→∞ 1 bn log P{Yn ∈ F } ≤ − inf λ∈F Λ∗(λ). If we further assume that the logarithmic moment function Λ(θ) is essentially smooth, then for any open set G ⊂ R, (1.1.5) lim inf n→∞ 1 bn log P{Yn ∈ G} ≥ − inf λ∈G Λ∗(λ). Proof. We first prove the upper bound (1.1.4). Without loss generality we may assume that inf λ∈F Λ∗(λ) 0. Let 0 l inf λ∈F Λ∗(λ) be fixed but arbitrary. By the convexity, lower semi-continuity and goodness of Λ∗(·) there are real numbers α β such that {λ ∈ R Λ∗(λ) ≤ l} = [α, β]. Clearly, F ∩ [α, β] = ∅. Therefore, there are α α and β β such that F ∩ (α , β ) = ∅. Obviously, Λ∗(α ), Λ∗(β ) l. We claim that Λ∗(α ) = sup θ0 θα − Λ(θ) , Λ∗(β ) = sup θ0 θβ − Λ(θ) . In fact, the relation Λ∗(α ) = sup θ≥0 θα − Λ(θ) would lead to Λ∗(α ) ≤ Λ∗(λ) for any λ α . This is impossible when λ ∈ [α, β]. In addition, P{Yn ∈ F } ≤ P{Yn ∈ (α , β )} = P{Yn ≤ α } + P{Yn ≥ β }. Consequently, lim sup n→∞ 1 bn log P{Yn ∈ F } ≤ max lim sup n→∞ 1 bn log P{Yn ≤ α }, lim sup n→∞ 1 bn log P{Yn ≥ β } . For any θ 0, P{Yn ≥ β } ≤ e−θβ bn E exp θbnYn . Hence, lim sup n→∞ 1 bn log P{Yn ≥ β } ≤ −θβ + Λ(θ). Taking infimum over θ 0 on the right hand side gives that lim sup n→∞ 1 bn log P{Yn ≥ β } ≤ −Λ∗(β ) ≤ −l.

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