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

Lutz Mattner, Uwe Rösler

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 sho…