Random Walk with Maximum and Minimum

Question

Do there exist stochastic walk process that are bounded? For example the value of the cumulative sum of the random values never exceeds some bound x, or goes lower than some y.

Answer 1

The only bounded random walk is the boring one where $S_n=0$ for all $n$. This follows from Theorem 4.1.2 (p. 155) of Probability: Theory and Examples (4th edition) by Richard Durrett. The book is freely available at the author's website.

Theorem 4.1.2. For a random walk on $\mathbb{R}$, there are only four possibilities, one of which has probability one.

1. $S_n=0$ for all $n$.

2. $S_n\to-\infty$.

3. $S_n\to\infty$.

4. $-\infty=\liminf S_n<\limsup S_n=\infty$.

There are lots of stochastic processes that are bounded, but they are not random walks.