Derivation, $L^ \Psi$-bounded martingales and covering conditions

Abstract:

Let $(\Omega , \Sigma , P)$ be a complete probability space. Let ${({\Sigma _t})_{t \in J}}$ be a directed family of sub-$\sigma$-algebras of $\Sigma$. Let $(\Phi , \Psi )$ be a pair of conjugate Young functions. We investigate the covering conditions that are equivalent to the essential convergence of ${L^\Psi }$-bounded martingales. We do not assume that either $\Phi$ or $\Psi$ satisfy the ${\Delta _2}$ condition. We show that when $\Phi$ satisfies condition Exp, that is when there exists an $a > 0$ such that $\Phi (u) \leq \operatorname {exp} au$ for each $u \ge 0$, the essential convergence of ${L^\Psi }$-bounded martingales is equivalent to the classical covering condition ${V_\Phi }$. This covers in particular the classical case $\Psi (t) = t{(\operatorname {log} t)^ + }$. The growth condition Exp on $\Phi$ cannot be relaxed. When $J$ contains a countable cofinite set, we show that the essential convergence of ${L^\Psi }$-bounded martingales is equivalent to a covering condition ${D_\Phi }$ (that is weaker than ${V_\Phi }$). When $\Phi$ fails condition Exp, condition ${D_\Phi }$ is optimal. Roughly speaking, in the case of ${L^1 }$-bounded martingales, condition ${D_\Phi }$ means that, locally, the Vitali condition with finite overlap holds. We also investigate the case where $J$ does not contain a countable cofinal set and $\Phi$ fails condition Exp. In this case, it seems impossible to characterize the essential convergence of ${L^\Psi }$-bounded martingales by a covering condition. Using the Continuum Hypothesis, we also produce an example where all equi-integrable ${L^1 }$-bounded martingales, but not all ${L^1 }$-bounded martingales, converge essentially. Similar results are also established in the derivation setting.