lecture-30 - Chapter 30 General Theory of Large Deviations...

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Chapter 30 General Theory of Large Deviations A family of random variables follows the large deviations princi- ple if the probability of the variables falling into “bad” sets, repre- senting large deviations from expectations, declines exponentially in some appropriate limit. Section 30.1 makes this precise, using some associated technical machinery, and explores a few consequences. The central one is Varadhan’s Lemma, for the asymptotic evalua- tion of exponential integrals in infinite-dimensional spaces. Having found one family of random variables which satisfy the large deviations principle, many other, related families do too. Sec- tion 30.2 lays out some ways in which this can happen. As the great forensic statistician C. Chan once remarked, “Improbable events permit themselves the luxury of occurring” (reported in Biggers, 1928). Large deviations theory, as I have said, studies these little luxuries. 30.1 Large Deviation Principles: Main Defini- tions and Generalities Some technicalities: Definition 397 (Level Sets) For any real-valued function f : Ξ R , the level sets are the inverse images of intervals from -∞ to c inclusive, i.e., all sets of the form { x Ξ : f ( x ) c } . Definition 398 (Lower Semi-Continuity) A real-valued function f : Ξ R is lower semi-continuous if x n x implies lim inf f ( x n ) f ( x ) . 206
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CHAPTER 30. LARGE DEVIATIONS: BASICS 207 Lemma 399 A function is lower semi-continuous i ff either of the following equivalent properties hold. i For all x Ξ , the infimum of f over increasingly small open balls centered at x approaches f ( x ) : lim δ 0 inf y : d ( y,x ) < δ f ( y ) = f ( x ) (30.1) ii f has closed level sets. Proof: A character-building exercise in real analysis, left to the reader. Lemma 400 A lower semi-continuous function attains its minimum on every non-empty compact set, i.e., if C is compact and = , there is an x C such that f ( x ) = inf y C f ( y ) . Proof: Another character-building exercise in real analysis. Definition 401 (Logarithmic Equivalence) Two sequences of positive real numbers a n and b n are logarithmically equivalent , a n b n , when lim n →∞ 1 n (log a n - log b n ) = 0 (30.2) Similarly, for continuous parameterizations by > 0 , a b when lim 0 (log a - log b ) = 0 (30.3) Lemma 402 (“Fastest rate wins”) For any two sequences of positive num- bers, ( a n + b n ) a n b n . Proof : A character-building exercise in elementary analysis. Definition 403 (Large Deviation Principle) A parameterized family of ran- dom variables, X , > 0 , taking values in a metric space Ξ with Borel σ -field X , obeys a large deviation principle with rate 1 / , or just obeys an LDP , when, for any set B X , - inf x int B J ( x ) lim inf 0 log P ( X B ) lim sup 0 log P ( X B ) ≤ - inf x cl B J ( x ) (30.4) for some non-negative function J : Ξ [0 , ] , its raw rate function . If J is lower semi-continuous, it is just a rate function . If J is lower semi-continuous and has compact level sets, it is a good rate function . 1 By a slight abuse of notation, we will write J ( B ) = inf x B J ( x ) .
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