1.4. NOTES AND COMMENTS 23 where I(λ) = ⎧ ⎨ ⎩ λ log λ − λ + 1 λ ≥ 0, +∞ λ 0. In particular, compute the limit lim t→∞ 1 t log P Nt/t ≥ λ for every λ 0. Comment. Exercise 1.4.2 serves as a counterexample at an attempt to weaken the conditions assumed in Theorem 1.2.1 and Theorem 1.2.4. In this example, the LDP given by (1.2.1)-(1.2.2) and the LDP defined by (1.2.3) have different rate functions, and none of them are strictly increasing on R+. Exercise 1.4.3. Prove the upper bound of Varadhan’s integral lemma given in part (2) of Theorem 1.1.6. Section 1.2. Much material in this section exists in some recent research papers instead of standard textbooks. Theorem 1.2.2 was essentially obtained in the paper by Bass, Chen and Rosen ([8]), Theorem 1.2.7 appeared in Chen ([27], [28]). Theorem 1.2.8 is due to K¨ onig and M¨ orters ([114]). A weaker version of Theorem 1.2.8 is also established in [114]). We put it into the following exercise. Exercise 1.4.4. Let Y ≥ 0 be a random variable such that lim sup m→∞ 1 m log 1 (m!)γ EY m ≤ κ for some γ 0 and κ ∈ R. Prove directly that lim sup t→∞ t−1/γ log P{Y ≥ t} ≤ −γe−κ/γ. Hint: You may use Chebyshev inequality and Stirling formula. We now consider the asymptotic behavior of the probability P{Y ≤ } ( → 0+). The study in this direction is often referred to as the problem on small ball prob- ability, since very often Y = ||X|| for some variable X taking values in a Banach space. We refer to the survey paper by Li and Shao ([133]) for an overview in this area. Generally speaking, large deviation principle and small ball probability deal with drastically different situations. One is for the probability that the random variables take large values and another is for the probability that the random vari- ables take small values. In the following exercise, we seek a small ball version of the G¨ artner-Ellis theorem.

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