{"paper":{"title":"Local limit theorem for the maximum of a random walk","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Johannes Kugler","submitted_at":"2014-03-28T13:39:52Z","abstract_excerpt":"Consider a family of $\\Delta$-latticed aperiodic random walks $\\{S^{(a)},0\\le a\\le a_0\\}$ with increments $X_i^{(a)}$ and non-positive drift $-a$. Suppose that $\\sup_{a\\le a_0}\\mathbf{E}[(X^{(a)})^2]<\\infty$ and $\\sup_{a\\le a_0}\\mathbf{E}[\\max\\{0,X^{(a)}\\}^{2+\\varepsilon}]<\\infty$ for some $\\varepsilon>0$. Assume that $X^{(a)}\\xrightarrow[]{w} X^{(0)}$ as $a\\to 0$ and denote by $M^{(a)}=\\max_{k\\ge 0} S_k^{(a)}$ the maximum of the random walk $S^{(a)}$. In this paper we provide the asymptotics of $\\mathbf{P}(M^{(a)}=y\\Delta)$ as $a\\to 0$ in the case, when $y\\to \\infty$ and $ay=O(1)$. This asymp"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7372","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}