22 1. BASICS ON LARGE DEVIATIONS for small t. This can be extended to arbitary t 0 by sub-additivity. 1.4. Notes and comments Section 1.1. The earliest recorded work in large deviation theory is due to Cram´ er ([41]) and was published in 1938. The literature on large deviations is massive and it is impossible to list even a small portion of it. We point to the fundamental roles played by Donsker and Varadhan, and Freidlin and Wentzell in the birth of the modern theory of large deviations. The idea that the limit of the logarithmic moment generating function decides the large deviation goes back to Cram´ er ([41]). It has been formulated by artner ([84]) and Ellis ([73]) into a general theorem later known as the artner-Ellis theorem (Theorem 1.1.4). There are many excellent book accounts available in the theory of large deviations. We mention here the books by Varadhan [159], Freidlin and Wentzell [80], Ellis [74], Stroock [156], Deuschel and Stroock [53], Bucklew [20], Dembo and Zeitouni [47], den Hollander [97], Feng and Kurtz [77]. Finally, we refer an interested reader to the recent survey by Varadhan [161] for an overview on the latest development in the area of large deviations. Most of the material in this section comes from the book by Dembo and Zeitouni ([47]). For the large deviations in infinite dimensional space, a challenging part is to establish the exponential tightness. Theorem 1.1.7 provides a practical way of examining the exponential tightness. This useful result is due to de Acosta ([1]). Exercise 1.4.1. Let {Yn} and {Zn} are two sequences of real random variables such that lim n→∞ 1 bn log P |Yn Zn| = −∞. Show that if Yn obeys the large deviation principle with the scale bn and the good rate function I(·), then the same large deviation principle holds for Zn. Exercise 1.4.2. Recall that a Poisson process Nt is a stochastic process taking non-integer values such that (1) N0 = 0, (2) For any t 0, P Nt = m = e−t tm m! m = 0, 1, · · · , (3) For any s, t R+ with s t, Nt −Ns is independent of {Nu 0 u s} and has same distribution as Nt−s. Prove the following LDP: For any closed set F R and open set G R, lim sup t→∞ 1 t log P Nt/t F inf λ∈F I(λ), lim inf t→∞ 1 t log P Nt/t G inf λ∈G I(λ)
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