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We consider a one-dimensional random walk $S_n$ with i.i.d. increments withzero mean and finite variance. We study the asymptotic expansion for the taildistribution $\mathbf P(\tau_x>n)$ of the first passage times$\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptoticexpansion for local probabilities $\mathbf P(S_n=x,\tau_0>n)$. Studying theasymptotic expansions we obtain a sequence of discrete polyharmonic functionsand obtain analogues of renewal theorem for them.

Expansions for random walks conditioned to stay positive