1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 13 where (1.2.9) Ip(λ) = sup θ0 θλ1/p − Λp(θ) λ 0. Proof. Replacing Yn by Yn 1/p in Theorem 1.2.4 completes the proof. In view of the Taylor expansion, (1.2.10) E exp θbnYn 1/p = ∞ m=0 θm m! bmEY n m/p n , one may attempt to estimate EYn m/p when establishing (1.2.7) by “standard” ap- proach becomes technically difficult. When 1/p is not integer, however, it is not very pleasant to deal with the (possibly) fractional power m/p. To resolve this problem, we introduce the following lemma. Lemma 1.2.6. Let p 0 be fixed and let Ψ: [0, ∞) −→ [0, ∞] be a non-decreasing lower semi-continuous function. Assume that the domain of Ψ has the form DΨ ≡ {θ Ψ(θ) ∞} = [0,a) where 0 a ≤ ∞, and that Ψ(θ) is continuous on DΨ. (1) The lower bound (1.2.11) lim inf n→∞ 1 bn log ∞ m=0 θm m! bn m EYn m 1/p ≥ Ψ(θ) θ 0 holds if and only if (1.2.12) lim inf n→∞ 1 bn log E exp θbnYn 1/p ≥ pΨ θ p θ 0. (2) The upper bound (1.2.13) lim sup n→∞ 1 bn log ∞ m=0 θm m! bn m EYn m 1/p ≤ Ψ(θ) θ 0 holds if and only if (1.2.14) lim sup n→∞ 1 bn log E exp θbnYn 1/p ≤ pΨ θ p θ 0. Proof. We first prove “(1.2.11) =⇒ (1.2.12)”. We may assume that in (1.2.12), the right hand side is positive. By the expansion (1.2.10) we have E exp θbnYn 1/p ≥ θ[pm]+1 ([pm] + 1)! b[pm]+1EYn n [pm]+1 p m = 0, 1, · · · . By Jensen inequality bn [pm]+1 EYn [pm]+1 p ≥ bn pm EYn m [pm]+1 pm . Therefore, as bn pm EYn m ≥ 1 we have bn [pm]+1 EYn [pm]+1 p ≥ bn pm EYn m .
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