Randomized Algorithms (Spring 2010)/Martingales

Martingales

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]

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, }[/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]