Applied Mathematicsematics

Large Deviations Techniques and Applications, Second Edition by Amir Dembo, Ofer Zeitouni

By Amir Dembo, Ofer Zeitouni

Huge deviation estimates have proved to be the an important device required to address many questions in records, engineering, statistial mechanics, and utilized likelihood. Amir Dembo and Ofer Zeitouni, of the major researchers within the box, offer an creation to the idea of huge deviations and functions at a degree appropriate for graduate scholars. the maths is rigorous and the functions come from a variety of parts, together with electric engineering and DNA sequences. the second one version, revealed in 1998, incorporated new fabric on focus inequalities and the metric and vulnerable convergence techniques to giant deviations. common statements and purposes have been sharpened, new workouts extra, and the bibliography up-to-date. the current delicate hide variation is a corrected printing of the 1998 variation.

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Extra info for Large Deviations Techniques and Applications, Second Edition

Example text

Wtid ), where wtij , i = 1, . . , n are samples of independent Brownian motions at time instances tj . √ Remark: Note that the law of Sˆn is the same as that of (1/ n) (wt1 , . . 3. d. case. d. case is possible. It is the goal of this section to present such an extension. 30 follows. 1 is devoted to a class of important applications, the large deviations of the empirical measure for finite state Markov chains. Consider a sequence of random vectors Zn ∈ IRd , where Zn possesses the law μn and logarithmic moment generating function Λn (λ)= log E e λ,Zn .

Ai ) > 0 while μ(ai ) = 0. 15). 16 Prove that for every open set Γ, − lim { inf n→∞ ν∈Γ∩Ln H(ν|μ)} = lim n→∞ 1 log Pμ (LY n ∈ Γ) n = − inf H(ν|μ)= − IΓ . 16 to any subset Γ of {ν ∈ M1 (Σ) : Σν ⊆ Σμ } that is contained in the closure of its interior. (b) Prove that for any such set, IΓ < ∞ and IΓ = H(ν ∗ |μ) for some ν ∗ ∈ Γo . Hint: Use the continuity of H(·|μ) on the compact set Γo . 19 Assume Σμ = Σ and that Γ is a convex subset of M1 (Σ) of non-empty interior. 18 apply. Moreover, prove that IΓ = H(ν ∗ |μ) for a unique ν ∗ ∈ Γo .

Yn ) = y) = e−nH(μ) . 8 For every ν ∈ Ln , (n + 1)−|Σ| enH(ν) ≤ |Tn (ν)| ≤ enH(ν) . /(nν(a1 ))! , a good estimate of |Tn (ν)| can be obtained from Stirling’s approximation. ) Here, a different route, with an information theory flavor, is taken. 1 Combinatorial Techniques for Finite Alphabets Proof: Under Pν , any type class has probability one at most and all its realizations are of equal probability. 7), −nH(ν) 1 ≥ Pν (LY |Tn (ν)| n = ν) = Pν ((Y1 , . . , Yn ) ∈ Tn (ν)) = e and the upper bound on |Tn (ν)| follows.

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