Abstract
dc:description.abstractWe consider a one-dimensional random walk S<sub>n</sub> with i.i.d. increments, zero mean and finite variance. Consider t<sub>x</sub> := inf{n ≥ 1 : x + S<sub>n</sub> ≤ 0} — the first passage times. For x ≥ 0 we study the asymptotic expansion for the tail distribution P(t<sub>x</sub> > n) under the condition that one-step distribution has finite high moments. We also derive asymptotic expansion for local probabilities P(S<sub>n</sub> = x, t<sub>0</sub> > 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.
Degree
thesis:*- Level thesis:degree_level
- thesis.doctoral
- Grantor dc:publisher
- Universität Bielefeld
- Year
- 2024
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Tarasov, Aleksandr
Identifiers
dc:identifier.*- Repository record source_url
- https://pub.uni-bielefeld.de/record/2999371
- OAI identifier oai:identifier
- oai:pub.uni-bielefeld.de:2999371