12 1. BASICS ON LARGE DEVIATIONS By Theorem 1.2.1 it remains to show that I(λ) is strictly increasing and con- tinuous on R+. From (1.2.5) and by the steepness of Λ(θ), I(λ) is well defined on R+, non-negative and non-decreasing. Let λ 0 be fixed. By essential smoothness of Λ(θ) and by (1.2.6), the function h(θ) = λθ − Λ(θ) θ ≥ 0 is bounded from above and attains its supremum I(λ) at some θ ≥ 0. Further, θ 0, for otherwise (observe that Λ(0) = 0) Λ(θ) ≥ λθ for all θ 0, which contradicts the assumption that Λ (0+) = 0 (again, we use the fact Λ(0) = 0 here). In summary, for any λ 0 there is a θ 0, such that Λ (θ) = λ and that I(λ) = λθ − Λ(θ). Let 0 λ1 λ2 a and find θ1 0 such that I(λ1) = λ1θ1 − Λ(θ1). We have I(λ2) ≥ θ1λ2 − Λ(θ1) θ1λ1 − Λ(θ1) = I(λ1). In addition, the above argument also leads to the fact that I(λ) 0 = I(0) for any λ 0. Hence, I(·) is strictly increasing on R+. Being increasing and lower semi-continuous, I(·) is left continuous. To establish continuity for I(λ) in R+, therefore, all we need to show is that for any 0 ≤ λ0 λn with λn → λ+, 0 I(λn) −→ I(λ0). Indeed, find θn 0 such that I(λn) = θnλn − Λ(θn) and Λ (θn) = λn n = 1, 2, · · · . In particular, the sequence {Λ (θn)} is bounded. By essential smoothness of Λ(·), {θn} is bounded. Hence 0 ≤ I(λn) − I(λ0) = θnλn − Λ(θn) − I(λ0) ≤ θn(λn − λ0) −→ 0 as n → ∞. The major step for establishing a large deviation principle by the G¨artner-Ellis theorem is to compute the exponential moment generating function Λ(θ) = lim n→∞ 1 bn log E exp θbnYn . When the exponential moment generating function is too diﬃcult to deal with, or when the exponential generating function blows up, we may look for some other moment functions instead. For example, we may consider the large deviations under the existence of the limit (1.2.7) Λp(θ) ≡ lim n→∞ 1 bn log E exp θbnYn 1/p θ ≥ 0 where p 0 is fixed. Corollary 1.2.5. Assume that for all θ 0, the limit Λp(θ) given in (1.2.7) exists as an extended real number, and that Λp(θ) is essentially smooth on R+ according to Definition 1.2.3. Then (1.2.8) lim n→∞ 1 bn log P{Yn ≥ λ} = −Ip(λ) λ 0

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