Consider a random walk on [0, inf) where you start at 0. With probability p = 0.5, you increase by 1. With probability (1-p) = 0.5, you decrease by 1, but not below 0.

As time goes to infinity, will your position tend to infinity? If not, to what finite value does it converge?

Edit: To be a bit more precise, what is the limit of the average position as time goes to infinity?

If your position tends to infinity for p = 0.5, for which other probabilities p is this true? (Clearly p > 0.5 will cause you to tend to infinity, so p < 0.5 is what I'm after)

What is the probability of being at position x after an arbitrary amount of steps?

I made a simple simulation to test the p = 0.5 case, and after 500 million iterations, it seems to tend to infinity, but I'd like a more solid explanation.

Thanks!