{"paper":{"title":"Higher Order Derivatives in Costa's Entropy Power Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Fan Cheng, Yanlin Geng","submitted_at":"2014-09-19T08:14:27Z","abstract_excerpt":"Let $X$ be an arbitrary continuous random variable and $Z$ be an independent Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa proved that $e^{2h(X+\\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on the first and second order derivatives of $h(X+\\sqrt{t}Z)$. Specifically, these two derivatives are signed, i.e., $\\frac{\\partial}{\\partial t}h(X+\\sqrt{t}Z) \\geq 0$ and $\\frac{\\partial^2}{\\partial t^2}h(X+\\sqrt{t}Z) \\leq 0$. In this paper, we show that the third order derivative of $h(X+\\sqrt{t}Z)$ is nonnegative, which implies that the Fisher information $J(X+\\sq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.5543","kind":"arxiv","version":3},"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"}