{"paper":{"title":"Karhunen-Lo\\`eve expansion for a generalization of Wiener bridge","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Matyas Barczy, Rezs\\H{o} L. Lovas","submitted_at":"2016-02-16T16:41:14Z","abstract_excerpt":"We derive a Karhunen-Lo\\`eve expansion of the Gauss process $B_t - g(t)\\int_0^1 g'(u)\\,d B_u$, $t\\in[0,1]$, where $(B_t)_{t\\in[0,1]}$ is a standard Wiener process and $g:[0,1]\\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\\int_0^1 (g'(u))^2\\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\\frac{\\sqrt{2}}{\\pi}\\sin(\\pi t)$, $t\\in[0,1]$, and $g(t)=t$, $t\\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.05084","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"}