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!



Will a random walk on [0, inf) tend to infinity?
The symmetric random walk $(X\_k)$ on $\mathbb{Z}$ is recurrent. Therefore, with probability one, you will visit infinitely often $0$. The same is true for $(|X\_k|)$, which is more ore less your random walk (the more or less depends on what happens exactly at the origin).


In other words, with probability one, the $\liminf$ is $0$. The $\limsup$ behaviour is the subject of the law of the iterated logarithm. 


