Randomized Algorithms (Spring 2010)/Martingales: Difference between revisions

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|'''Azuma's Inequality:'''  
|'''Corollary:'''  
:Let <math>X_0,X_1,\ldots</math> be a martingale such that, for all <math>k\ge 1</math>,
:Let <math>X_0,X_1,\ldots</math> be a martingale such that, for all <math>k\ge 1</math>,
::<math>
::<math>

Revision as of 07:44, 6 April 2010

Martingales

Review of conditional probability

Martingales and Azuma's Inequality

Azuma's Inequality:
Let [math]\displaystyle{ X_0,X_1,\ldots }[/math] be a martingale such that, for all [math]\displaystyle{ k\ge 1 }[/math],
[math]\displaystyle{ |X_{k}-X_{k-1}|\le c_k, }[/math]

Then

[math]\displaystyle{ \begin{align} \Pr\left[|X_n-X_0|\ge t\right]\le 2\exp\left(-\frac{t^2}{2\sum_{k=1}^nc_k^2}\right). \end{align} }[/math]
Corollary:
Let [math]\displaystyle{ X_0,X_1,\ldots }[/math] be a martingale such that, for all [math]\displaystyle{ k\ge 1 }[/math],
[math]\displaystyle{ |X_{k}-X_{k-1}|\le c, }[/math]

Then

[math]\displaystyle{ \begin{align} \Pr\left[|X_n-X_0|\ge ct\sqrt{n}\right]\le 2 e^{-t^2/2}. \end{align} }[/math]

Generalizations

The Method of Bounded Differences

Applications