The Simple Symmetric Random Walk

The simple random walk describes walking along the x – axis, starting at the origin and randomly moving to left and right, one space at a time.

Definition

A stochastic processwithis called a simple symmetric random walk if

1. the incrementis independent ofandfor each

2. the incrementhas the “coin toss distribution”

We can define a random variableto take the continuous uniform distribution onwith generatingaccording to the following rule

Then setand

Proving this satisfies the requirements of a random symmetric walk is quite easy. 1. is trivially true. To prove 2 note thatis an independent sequence since it is constructed by application of a deterministic function to each element of an independent sequencethenis independent of all previousSince theare linear combinations of thethey must also be independent ofFinally to obtain 3 note thatand similarly for