1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 15 We now prove “(1.2.12) =⇒ (1.2.11)”. We may assume that the right hand side of (1.2.11) is positive. For any k 0, θ[p−1k]+1 ([p−1k] + 1)! bn [p−1k]+1 EYnp−1k]+1 [ 1/p m=0 θm m! bn m EYn m 1/p . By Jensen inequality, Stirling formula and an argument similar to the one used for (1.2.15), we can prove that for any 0 δ there is C 0 independent of n such that E exp 1 + bnYn 1/p (1.2.17) exp 1 + + C 1 + δ δ m=0 θm m! bm n EY m n 1/p p . By (1.2.12) (with θ being replaced by 1 + ), we get θ 1 + max 0, p lim inf n→∞ 1 bn log m=0 θm m! bn m EYn m 1/p . Since the left hand side is positive, we have lim inf n→∞ 1 bn log m=0 θm m! bm n EY m n 1/p Ψ θ 1 + . By the lower semi-continuity of Ψ(·), letting 0+ on the right hand side gives (1.2.11). Finally, “(1.2.13) =⇒ (1.2.14)” also follows from the estimate given in (1.2.17). An immediate application of Lemma 1.2.6 is the following G¨artener-Ellis-type theorem. Theorem 1.2.7. Assume that for each θ 0, the limit (1.2.18) Ψ(θ) lim n→∞ 1 bn log m=0 θm m! bn m EYnm 1/p exists as an extended real number. Assume that the function Ψ(θ) is essentially smooth on R+ (Definition 1.2.3). For each λ 0, (1.2.19) lim n→∞ 1 bn log P{Yn λ} = −IΨ(λ) where the rate function IΨ(·) is defined by (1.2.20) IΨ(λ) = p sup θ0 θλ1/p Ψ(θ) λ 0. Proof. By Lemma 1.2.6, the condition posed in Corollary 1.2.5 is satisfied by Λp(θ) = pΨ(θ/p). We now consider the case of a single random variable.
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