{"paper":{"title":"Area distribution and the average shape of a L\\'evy bridge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"cond-mat.stat-mech","authors_text":"Gregory Schehr, Satya N. Majumdar","submitted_at":"2010-04-28T14:39:17Z","abstract_excerpt":"We consider a one dimensional L\\'evy bridge x_B of length n and index 0 < \\alpha < 2, i.e. a L\\'evy random walk constrained to start and end at the origin after n time steps, x_B(0) = x_B(n)=0. We compute the distribution P_B(A,n) of the area A = \\sum_{m=1}^n x_B(m) under such a L\\'evy bridge and show that, for large n, it has the scaling form P_B(A,n) \\sim n^{-1-1/\\alpha} F_\\alpha(A/n^{1+1/\\alpha}), with the asymptotic behavior F_\\alpha(Y) \\sim Y^{-2(1+\\alpha)} for large Y. For \\alpha=1, we obtain an explicit expression of F_1(Y) in terms of elementary functions.  We also compute the average "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.5046","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"}