24 1. BASICS ON LARGE DEVIATIONS Exercise 1.4.5 (hard). Let Y ≥ 0 be a random variable such that lim t→∞ 1 t log E exp − tγY = −κ for some γ 1 and κ 0. Prove that lim →0+ 1 γ−1 log P{Y ≤ } = −(γ − 1)γ− γ γ−1 κ γ γ−1 . (1.4.1) This result is known as exponential Tauberian theorem (see Theorem 3.5, [133] for a more general form). Hint: Use Chebyshev inequality to get the upper bound. The lower bound can be proved in a way similar to some argument used in the proof of Theorem 1.1.4. In particular, you may need to prove that for any interval I ⊂ R+ and θ 0, lim sup →0+ 1 γ−1 log E exp − θ− γ γ−1 Y 1{Y ∈I} ≤ − inf λ∈I θλ + (γ − 1)γ− γ γ−1 κ γ γ−1 λ− 1 γ−1 . Section 1.3. Argument by sub-additivity has become a sophisticated tool in the general frame work of large deviations. Very often in literature, it is the deterministic sub-additivity formulated in Lemma 1.3.1 that is used to prove the existence of the logarithmic moment generating function. Exercise 1.4.6. Prove (1.3.7).
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