PUB - Publikationen an der Universität Bielefeld

We consider a one-dimensional random walk S_n with i.i.d. increments, zero mean and finite variance. Consider t_x := inf{n ≥ 1 : x + S_n ≤ 0} — the first passage times. For x ≥ 0 we study the asymptotic expansion for the tail distribution P(t_x > n) under the condition that one-step distribution has finite high moments. We also derive asymptotic expansion for local probabilities P(S_n = x, t₀ > n). The cases of lower deviations (x=o(√n)) and of normal deviations (x ∼ √n) considered separately, they require different approaches and lead to different answers, moreover the proof for x ∼ √n is based on asymptotic expansions for smaller x. Studying the asymptotic expansions in lower deviation case we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.