Optimal L$^1$-bounds for submartingales

TL;DR

This paper derives the optimal L^1 bounds for submartingales and martingales, providing explicit formulas and demonstrating their optimality through convex-analytic methods.

Contribution

It introduces explicit formulas for the optimal bounds of submartingales and martingales, advancing understanding of their L^1 behavior with novel convex analysis techniques.

Findings

Explicit formula for optimal L^1 bounds for martingales.

Extension of bounds to submartingales.

Use of convex-analytic comparison lemma for proofs.

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 $$ f(a) = \max \Big\{a_k-\sum_{i=1}^{k-1} a_i\Big\}_{k=1}^n \cup \Big\{\frac {a_k}2\Big\}_{k=3}^n $$ 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.