I need to describe some properties of the random walk on numbers $0, 1, 2, \cdots$. So, this process bounded on the left side and unbounded on the right. Transitions possible only to adjacent numbers, and for zero - left neighbour is zero itself. Probabilities of transition to the left and to the right equal to $\frac{1}{2}$.
I need to calculate expectation of position if start position equals to $0$. I interested in both cases - expectation after $n$ steps and expectation after infinity steps.
Let me give you an example:
After 0 step: $0$ with prob $1.0$
After 1 step: $0$ with prob $0.5$, $1$ with prob $0.5$
After 2 steps: $0$ with prob $0.5$, $1$ with prob $0.25$, $2$ with prob $0.25$
etc.
Can you suggest some ideas, articles about this topic? Because I found many materials on the Internet about a random walk on infinity line, but none about a walk on an infinite ray.