Suppose that a gambler with initial capital $a$ is betting against an opponent with infinite capital in a fair game. Given that the gambler is NOT yet ruined until time $n$, what is the expected capital of the gambler at time $n$?
I have an asymptotic argument: the random walk conditioned to to stay positive until time $n$, when scaled in the usual diffusive way, should converge to the Brownian Meander process. What we are looking at is value at time $1$ after scaling. So, the asymptotic limit of expected value of the walk at time $n$, when scaled by $ \sqrt{n} $, should converge to a constant, namely the expected value of the meander at time $1$. This argument should hold for general random variables.
I am wondering if any specific formula is known when it is a simple symmetric random walk. Even a lower bound should be OK for my purpose. Any ideas or suggestions are most welcome.