Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints

We consider a one-dimensional random walk $S_n$ with i.i.d. integer-valued increments having zero mean and finite variance. Under suitable moment assumptions, we derive asymptotic expansions for the local probabilities $\pr(x+S_n=y,τ_x>n)$, where $τ_x=\inf\{n\ge1:x+S_n\le0\}$. These extend previous