Multi-Page Printing

1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 9 (2) For every λ 0, (1.2.3) lim n→∞ 1 bn log P Yn ≥ λ = −I(λ). Proof. Assume that (1) holds. Taking F = [λ, ∞) in (1.2.1) gives that lim sup n→∞ 1 bn log P Yn ≥ λ ≤ − inf x≥λ I(x) = −I(λ). Similarly, by (1.2.2) with G = (λ, ∞) we get lim inf n→∞ 1 bn log P Yn ≥ λ ≥ − inf xλ I(x) = −I(λ) where the last step follows from the continuity. Assume that (2) holds. For a closed set F ⊂ R+, let λ0 = inf F . We have that P{Yn ∈ F } ≤ P{Yn ≥ λ0}. By (1.2.3) we have lim sup n→∞ 1 bn log P Yn ∈ F ≤ −I(λ0) = − inf λ∈F I(λ) where the last step follows from the monotonicity of I(·). To establish (1.2.2), let λ0 ∈ G. There is a δ 0 such that [λ0,λ0 + δ) ⊂ G. By the fact that P{Yn ≥ λ0} ≤ P{Yn ≥ λ0 + δ} + P{Yn ∈ [λ0,λ0 + δ)} and by (1.2.3) we have −I(λ0) ≤ max − I(λ0 + δ), lim inf n→∞ 1 bn log P{Yn ∈ [λ0,λ0 + δ)} . By assumption we have that I(λ0 + δ) I(λ0). Consequently, lim inf n→∞ 1 bn log P Yn ∈ G ≥ −I(λ0). Taking supremum over λ0 ∈ G on the right hand side gives (1.2.2). The following theorem shows that under certain conditions, the tail probability of the fixed sum of independent non-negative random variables is dominated by the tail probabilities of individual terms. Theorem 1.2.2. Let Z1(n), · · · , Zl(n) be independent non-negative random vari- ables with l ≥ 2 being fixed. (a) If there are constant C1 0 and 0 a ≤ 1 such that lim sup n→∞ 1 bn log P Zj(n) ≥ λ ≤ −C1λa ∀λ 0 for j = 1, · · · , l, then lim sup n→∞ 1 bn log P Z1(n) + · · · + Zl(n) ≥ λ ≤ −C1λa ∀λ 0.