Introduction to Probability · Problem 12.1.15

Random Walks — Grinstead & Snell

Mathematics — Probability · random-walks-higher-dimensions

Problem

Using the result proved for the random walk in

\mathbf{R}^3

in Example 12.2, explain why the probability of an eventual return in

\mathbf{R}^n

is strictly less than one, for all

n \geq 3

. Hint: Consider a random walk in

\mathbf{R}^n

and disregard all but the first three coordinates of the particle's position.