Probability · Maths EE idea · Ambitious
Does a random walk come home?
A research question to start from
Why does a simple random walk return to its start with probability 1 in one and two dimensions but not in three?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
A famous dichotomy with a counting proof and series convergence at its heart.
Mathematics you would need
- Counting paths with binomial coefficients
- Stirling's approximation
- Convergence of series
- Expected number of returns
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Count returning paths in one dimension.
- Link recurrence to the divergence of the expected number of returns.
- Estimate terms in two and three dimensions and conclude.
Scope and difficulty
Ambitious. Ambitious; the link between returns and the series must be explained clearly.
Pitfalls
- Simulation instead of proof.
- Hand-waving the 3D estimate.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Polya recurrence theorem random walk proof; expected returns series. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).