Random Walk Probability

Question

Consider the infinite random walk on the integers that goes from i to i+1 in one move with probability 4/7, goes from i to i-1 in one move with probability 1/7, and finally goes from i to i-2 in one move with probability 2/7, Starting from 0, what percentage of the negative integers are NOT visited by this random walk? give a numerical value

Answer 1

Hi Jon K.,

Since your average displacement per turn is -1/7, over an infinite time you will visit all negative integers, guaranteed. For that matter, you will also visit all the positive integers. Infinities are funny that way.

Your expected position after many turns is expected to approach a normal distribution, which will spread out gradually as well as drift left gradually. It spreads out faster than it drifts left (like oil on water!), so it can eventually reach all integers on (-infinity, +infinity).

A more useful type of question here might be the probabilities of hitting particular integers within specified finite times. That's certainly complicated by the odd/even shifting action. I don't know how to approximate that in closed form; you might have fun setting it up as a Monte Carlo thing, and watching your integer bins fill! I have done this in the past, with an Excel macro, but there are surely better platforms available now.

--Cheers, Mr. d.