lecture-4

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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online