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

Revision as of 07:37, 6 April 2010

Azuma's Inequality:

Let [math]\displaystyle{ X_0,X_1,\ldots }[/math] be a martingale such that for each [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]

@@ Line 4: / Line 4: @@
 === Martingales and Azuma's Inequality ===
+{|border="1"
+|'''Azuma's Inequality:'''
+:Let <math>X_0,X_1,\ldots</math> be a martingale such that for each <math>k\ge 1</math>,
+::<math>
+|X_{k}-X_{k-1}|\le c_k,
+</math>
+Then
+::<math>\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>
+|}
 === Generalizations ===