Introduction to Probability · Problem 12.1.8

Random Walks — Grinstead & Snell

Mathematics — Probability · random-walks-positive-paths

Problem

Show that

P(S_1 \geq 0,\, S_2 \geq 0,\, \ldots,\, S_{2m} \geq 0) = u_{2m}.

Hint: First explain why

P(S_1 > 0,\, S_2 > 0,\, \ldots,\, S_{2m} > 0) = \frac{1}{2} P(S_1 \neq 0,\, S_2 \neq 0,\, \ldots,\, S_{2m} \neq 0).

Then use Exercise 7, together with the observation that if no equalization occurs in the first

2m

outcomes, then the path goes through the point

(1,1)

and remains on or above the horizontal line

x = 1