A type of stochastic jump process Let $X \geq 1$ be an integer r.v. with $E[X]=\mu$. Let $X_i$ be a sequence of iid rvs with the distribution of $X$. On the integer line, we start at $0$, and want to know the expected position after we first cross $K$, which is some fixed integer. Each next position is determined by adding $X_i$ to the previous position. So the question is, if we stop this process after the first time $\tau$ for which $Y_{\tau}=\sum_{i=1}^{\tau}X_i > K$, that is, after the first time it crosses $K$, then what is $E[Y_{\tau}-K]$?.   Can we get a bound of $O(\mu)$?

Edit: This answer shows what happens when $X$ could be zero (contrary to the question).
Let $K = 1$, and $X = M$ ($M > 1$) with probability $p$ and $0$ otherwise. Thus $E[Y_\tau - K] = M-1$ whereas $E[X] = pM$. Choosing $M = n+1$ and $p = 1/n(n+1)$ we get $E[X] = 1/n$ and $E[Y_\tau - K] = n$, so a bound of $O(\mu)$ isn't possible.
