Skip to main content

TL;DR

Return Probability of Random Walks by Dimension: A canonical quantitative trading interview question at olympiad difficulty. Commonly asked at Jane Street, Two Sigma, DE Shaw.

By Valenke Exam Prep Team·Last updated 2026-06-01
olympiadMarkov Chains & Random Walks

Return Probability of Random Walks by Dimension

Asked at: Jane Street, Two Sigma, DE Shaw

Problem
A symmetric random walk on Zd\mathbb{Z}^d starts at the origin. In 1D and 2D, the walk returns to the origin with probability 1 (Pólya's theorem). In 3D, the return probability is strictly less than 1. Compute the return probability for d=3d = 3.

Ready to practice for the Valenke Finance Exam?

Adaptive practice powered by Item Response Theory targets your weak areas. Start with 3 free sessions.

Start free practice →