lecture-34 - Chapter 34 Large Deviations for Weakly Dep...

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Chapter 34 Large Deviations for Weakly Dependent Sequences: The artner-Ellis Theorem This chapter proves the G¨ artner-Ellis theorem, establishing an LDP for not-too-dependent processes taking values in topological vector spaces. Most of our earlier LDP results can be seen as con- sequences of this theorem. 34.1 The G¨ artner-Ellis Theorem The G¨ artner-Ellis theorem is a powerful result which establishes the existence of a large deviation principle for processes where the cumulant generating func- tion tends towards a well-behaved limit, implying not-too-strong dependence between successive values. (Exercise 34.5 clarifies the meaning of “too strong”.) It will imply our LDPs for IID and Markovian sequences. I could have started with it, but its proof, as you’ll see, is pretty technical, and so it seemed better to use the more elementary arguments of the preceding chapters. To fix notation, Ξ will be a real topological vector space, and Ξ * will be its dual space, of continuous linear functions Ξ ±→ R . (If Ξ = R d , we can identify Ξ and Ξ * by means of the inner product. In diFerential geometry, on the other hand, Ξ might be a space of tangent vectors, and Ξ * the corresponding one- forms.) X ± will be a family of Ξ-valued random variables, parameterized by ± > 0. Refer to Definitions 423 and 424 in Section 31.1 for the definition of the cumulant generating function and its Legendre transform (respectively), which I will denote by Λ ± : Ξ * ±→ R and Λ * ± : Ξ ±→ R . The proof of the G¨ artner-Ellis theorem goes through a number of lemmas. 233
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CHAPTER 34. THE G ¨ ARTNER-ELLIS THEOREM 234 Basically, the upper large deviation bound holds under substantially weaker conditions than the lower bound does, and it’s worthwhile having the partial results available to use in estimates even if the full large deviations principle does not apply. Definition 444 The upper-limiting cumulant generating function is Λ( t ) lim sup ± 0 ± Λ ± ( t/± ) (34.1) and its Legendre transform is written Λ * ( x ) . The point of this is that the limsup always exists, whereas the limit doesn’t, necessarily. But we can show that the limsup has some reasonable properties, and in fact it’s enough to give us an upper bound. Lemma 445
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lecture-34 - Chapter 34 Large Deviations for Weakly Dep...

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