1.2. LDP FOR NON-NEGATIVE RANDOM VARIABLES 17 (1) Assume that (1.2.23) lim inf n→∞ 1 bn log P{Yn ≥ λ} ≥ −I(λ) (λ 0). Then for every θ 0 (1.2.24) lim inf n→∞ 1 bn log ∞ m=0 θm m! bn m EYn m 1/p ≥ sup λ0 θλ1/p − p−1I(λ) . (2) Assume that (1.2.25) lim sup n→∞ 1 bn log P{Yn ≥ λ} ≤ −I(λ) (λ 0). Then for every θ 0 satisfying (1.2.26) lim l→∞ lim sup n→∞ 1 bn log ∞ m=0 θm m! bm n E ( Y m n 1{Y n ≥l} ) 1/p = 0 we have (1.2.27) lim sup n→∞ 1 bn log ∞ m=0 θm m! bn m EYn m 1/p ≤ sup λ0 θλ1/p − p−1I(λ) . Proof. For any λ 0, EY m n ≥ λmP{Yn ≥ λ}. Consequently, ∞ m=0 θm m! bm n EY m n 1/p ≥ exp bnθλ1/p P{Yn ≥ λ} 1/p . By (1.2.23) lim inf n→∞ 1 bn log ∞ m=0 θm m! bn m EYn m 1/p ≥ θλ1/p − p−1I(λ). Taking supremum over λ 0 on the right hand side gives (1.2.24). To prove (1.2.27), write ∞ m=0 θm m! bm n EY m n 1/p (1.2.28) = ∞ m=0 θm m! bm n E ( Y m n 1{Y n l} ) 1/p + ∞ m=0 θm m! bm n E ( Y m n 1{Y n ≥l} ) 1/p . Given 0, partition the interval [0,l] into 0 = λ0 · · · λN = l such that for each 1 ≤ i ≤ l the length of the sub-interval Ai = [λi−1,λi] is less than . So
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