{"paper":{"title":"Small deviations in lognormal Mandelbrot cascades","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Miika Nikula","submitted_at":"2013-06-14T16:34:11Z","abstract_excerpt":"We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if $\\log \\log 1/P(W \\leq x) \\sim \\gamma \\log \\log 1/x$ as $x \\to 0$ with $\\gamma > 1$, then the Laplace transform of $Y$ satisfies $\\log \\log 1/\\E e^{-t Y} \\sim \\gamma \\log \\log t$ as $t \\to \\infty$. As an application, this gives new estimates for $\\Prob(Y \\leq x)$ for small $x > 0$. As another application of our methods, we prove a similar result for a variable arising as a total mass of a lognormal $\\star$-scale invariant mul"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1306.3448","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"}