{"paper":{"title":"Exponential Family Techniques for the Lognormal Left Tail","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jens Ledet Jensen, Leonardo Rojas-Nandayapa, Soren Asmussen","submitted_at":"2014-03-19T04:36:32Z","abstract_excerpt":"Let $X$ be lognormal$(\\mu,\\sigma^2)$ with density $f(x)$, let $\\theta>0$ and define ${L}(\\theta)=E e^{-\\theta X}$. We study properties of the exponentially tilted density (Esscher transform) $f_\\theta(x) =e^{-\\theta x}f(x)/{L}(\\theta)$, in particular its moments, its asymptotic form as $\\theta\\to\\infty$ and asymptotics for the Cram\\'er function; the asymptotic formulas involve the Lambert W function. This is used to provide two different numerical methods for evaluating the left tail probability of lognormal sum $S_n=X_1+\\cdots+X_n$: a saddlepoint approximation and an exponential twisting impo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.4689","kind":"arxiv","version":1},"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"}