Optimal L$^1$-bounds for submartingales
Abstract
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.
14 pages. Minor corrections and notational changes. Address of first-named author updated
14 pages. Minor corrections and notational changes. Address of first-named author updated