Optimal L${}^1$-bounds for submartingales

Abstract

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 $$f(a)=\max \{a_k-\sum _{i=1}^{k-1}{a}_i{\}}_{k=1}^n\cup \{\frac{a_k}{2}{\}}_{k=3}^n$$ 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

References in corpus (1)

• Sums of norm spheres are norm shells and lower triangle inequalities are sharp