{"id":{"repo_id":"bielefeld","oai_identifier":"oai:pub.uni-bielefeld.de:2999371"},"canonical_url":"https://search.dev.ndltd.org/etd/bielefeld/oai:pub.uni-bielefeld.de:2999371","repository":{"repo_id":"bielefeld","name":"Universität Bielefeld","base_url":"https://pub.uni-bielefeld.de/oai"},"display":{"title":"Random walks conditioned to stay positive","abstract":"We 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.","abstract_html":"We consider a one-dimensional random walk S&lt;sub&gt;n&lt;/sub&gt; with i.i.d. increments, zero mean and finite variance. Consider t&lt;sub&gt;x&lt;/sub&gt; := inf{n ≥ 1 : x + S&lt;sub&gt;n&lt;/sub&gt; ≤ 0} — the first passage times. For x ≥ 0 we study the asymptotic expansion for the tail distribution P(t&lt;sub&gt;x&lt;/sub&gt; &gt; n) under the condition that one-step distribution has finite high moments. We also derive asymptotic expansion for local probabilities P(S&lt;sub&gt;n&lt;/sub&gt; = x, t&lt;sub&gt;0&lt;/sub&gt; &gt; 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.","abstract_has_math":false,"creators":["Tarasov, Aleksandr"],"institution":"Universität Bielefeld","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-11-20","date_published":"2024-11-20","updated_at":"2026-07-27T18:50:01Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://pub.uni-bielefeld.de/record/2999371","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Tarasov, Aleksandr"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universitätsbibliothek Bielefeld"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Bielefeld"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["We 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."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Random walks conditioned to stay positive"]}]}],"canonical_facts":{"dc:creator":["Tarasov, Aleksandr"],"dc:description.abstract":["We 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."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universitätsbibliothek Bielefeld"],"dc:title":["Random walks conditioned to stay positive"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Bielefeld"]},"updated_at":"2026-07-27T18:50:01Z"}