Show Random Walk Respects Identity

1. Apr 13, 2017

Polo Lagrie

• Moved thread to homework forum
Hi,

I have the following hw question:

Let Xt be the continuous-time simple random walk on a circle as in Example 2, Section 7.2. Show that there exists a c,β > 0, independent of N such that for all initial probability distributions ν and all t > 0

$$∥νe^tA−π∥_TV ≤ ce^(−βt/N2)$$

Here's what I'm thinking so far:

We proved the following in class:

$$∥νe^tA−π∥_TV ≤ 1/2 ||e^(tA) v/π −1||_π$$

I know I can apply this to the function above in the form:

$$∥νe^(tA) −π∥TV ≤1/2e^λ2t ||v/π||_π$$

But I'm now stuck. Any help would be greatly appreciated!

P.s. Sorry for the poor formatting--new at Latex.

2. Apr 19, 2017

PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.