Optimal L$^1$-bounds for submartingales
| dc.creator | Mattner, Lutz | |
| dc.creator | Rösler, Uwe | |
| dc.date | 2008-09-20 | |
| dc.date | 2009-04-16 | |
| dc.date.accessioned | 2026-07-07T13:04:04Z | |
| dc.date.available | 2026-07-07T13:04:04Z | |
| dc.description | The optimal function $f$ satisfying $$ \mathbb{E} |\sum_{1}^n X_i | \ge f(\mathrbb{E}|X_1|,...,\mathbb{E}|X_n|) $$ for every martingale $(X_1,X_1+X_2, ...,\sum_{i=1}^n X_i)$ is shown to be given by $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup \Big\{\frac {a_k}2\Big\}_{k=3}^n $$ for $a\in{[0,\infty[}^n_{}$. A similar result is obtained for submartingales $(0,X_1,X_1+X_2,..., \sum_{i=1}^n X_i)$. The optimality proofs use a convex-analytic comparison lemma of independent interest. | |
| dc.description | 14 pages. Minor corrections and notational changes. Address of first-named author updated | |
| dc.identifier | https://arxiv.org/abs/0809.3522 | |
| dc.identifier | http://arxiv.org/abs/0809.3522 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/227030 | |
| dc.subject | Probability | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | 60G42; 60E15; 26B25 | |
| dc.title | Optimal L$^1$-bounds for submartingales | |
| dc.type | text |