1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 11 A basic assumption of the artner-Ellis theorem (Theorem 1.2.4 below) for non- negative random variables is the existence of the logarithmic moment generating function on R+. Correspondent to Definition 1.1.3, we adopt the following notion of essential smoothness for the functions defined on R+. Definition 1.2.3. A convex function Λ(θ): R+ −→ [0, ∞] is said to be essen- tially smooth on R+, if (1) There is a θ0 0 such that Λ(θ) for every 0 θ θ0. (2) Λ(θ) is differentiable in the interior Do Λ = (0,a) (0 a ∞) of the domain = R+ Λ(θ) ∞}. (3) The function Λ(θ) is steep at the right end (of the domain) and is flat at the left end, i.e., lim θ→a− Λ (θ) = and Λ (0+) lim θ→0+ Λ(θ) Λ(0) θ = 0. The following theorem appears as a version of Ga¨ rtner-Ellis large deviation. Theorem 1.2.4. Assume that for all θ 0, the limit (1.2.4) Λ(θ) = lim n→∞ 1 bn log E exp θbnYn exists as an extended real number, and that the function Λ(θ) is essentially smooth on R+. (1) The function (1.2.5) I(λ) = sup θ0 θλ Λ(θ) λ 0 is strictly increasing and continuous on R+. Consequently, the LDP given defined by (1.2.1) and (1.2.2) and the LDP defined by (1.2.3) are equiva- lent. (2) The equivalent forms (1.2.1), (1.2.2) and (1.2.3) hold. Proof. Let ξ be an independent random variable with distribution P{ξ = −1} = P{ξ = 1} = 1 2 . We have E exp θbnξYn = 1 2 E exp θbnYn + E exp θbnYn θ R. Consequently, lim n→∞ 1 bn log E exp θbnξYn = Λ(|θ|) θ R. Notice that Λ(|θ|) is essentially smooth on R according to Definition 1.1.3. Applying Theorem 1.1.4 to the sequence {ξYn} we have the large deviation principle given in (1.2.1) and (1.2.2) with the rate function given by I(λ) = sup θ∈R θλ Λ(|θ|) = sup θ0 θλ Λ(θ) ∀λ 0. (1.2.6)
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