lecture-4

lecture-4 - EE378 Statistical Signal Processing Lecture 4...

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Unformatted text preview: EE378 Statistical Signal Processing Lecture 4 - 04/16/2007 Martingales, Hoeffding-Azuma Inequality Lecturer: Tsachy Weissman Scribe: Hattie Dong, Jia-Yu Chen, Kamakshi S 1 Martingales Definition 1 (Martingale) . Z 1 , Z 2 , ... is a martingale if E Z i +1 | Z i = Z i ∀ i , and a martingale w.r.t. X 1 , X 2 , ... if ∀ i :- Z i = f i ( X i )- E Z i +1 | X i = Z i As shown in the Martingale convergence theorem (Theorem 4 in Lecture Notes 3), for Z i ’s with bounded first moments, they converge w.p. 1. We will now prove a concentration inequality for a normalized martin- gale. Toward this end, we first define a Martingale difference sequence. Definition 2 (Martingale Difference) . V 1 , V 2 , ... is a martingale difference sequence if E V i +1 | V i = 0 ∀ i , and (more generally) a martingale difference sequence w.r.t. X 1 , X 2 , ... if ∀ i :- V i = g i ( X i )- E V i +1 | X i = 0 Notes: 1) (Like for martingales) V 1 , V 2 , ... is a martingale difference sequence w.r.t. X 1 , X 2 , ... ⇒ V 1 , V 2 , ... is a martingale difference sequence. 2) Z 1 , Z 2 , ... is a martingale ⇒ V i = Z i- Z i- 1 is a martingale difference sequence....
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lecture-4 - EE378 Statistical Signal Processing Lecture 4...

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