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

Abstract:

The optimal function $f$ satisfying \[ \mathbb {E} |\sum _{1}^n X_i | \ge f(\mathbb {E}|X_1|,\ldots ,\mathbb {E}|X_n|) \] for every martingale $(X_1,X_1+X_2, \ldots ,\sum _{i=1}^n X_i)$ 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 $(0,X_1, X_1+X_2,\ldots , \sum _{i=1}^n X_i)$.

The optimality proofs use a convex-analytic comparison lemma of independent interest.