1 Random Walks
1.1 What is a random walk?
Let \(\{\xi_i\}_{i=1}^\infty\) be a sequence of i.i.d. random variables. A random walk is a stochastic process \(\{S_n\}_{n=1}^\infty\) defined recursively by
\[ S_n = S_{n-1}+\xi_n. \]
Through repeated substitution we obtain the closed form expression
\[ S_n = S_0 + \sum_{i=1}^n \xi_i. \]
If the step process \(\{\xi_i\}\) has step size 1 with mass function
\[ \mathbb{P}(\xi_i=x) = \begin{cases} p & \text{if }x=1; \\ 1-p & \text{if } x=-1, \end{cases} \]
then we say that the process is simple. Furthermore, if \(p=1/2\) we say that the process is symmetric.