{"paper":{"title":"The LIL for canonical U-statistics of order 2","license":"","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Evarist Gin\\'e, Joel Zinn, Rafa{\\l} Lata{\\l}a, Stanis{\\l}aw Kwapie\\'n","submitted_at":"1999-05-20T17:19:17Z","abstract_excerpt":"Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\\limsup_n (n\\log\\log n)^{-1}|\\sum_{1<= i< j<= n}h(X_i,X_j)|<\\infty$ a.s., holds if and only if the following three conditions are satisfied: h is canonical for the law of X (that is Eh(X,y)=0 for almost y) and there exists $C<\\infty$ such that, both, $E\\min(h^2(X_1,X_2),u)<C\\log\\log u$ for all large u and $sup\\{Eh(X_1,X_2)f(X_1)g(X_2):|f(X)|_2<1,\\|g(X)\\|_2<1, \\|f\\|_\\infty<\\infty, \\|g\\|_\\infty<\\inf"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9905132","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"}