Some Structure Results for Martingales in the Limit and Pramarts

Abstract

We show that an $L^1$-bounded Banach-space-valued martingale in the limit $(X_n)$ can be written $X_n = Y_n + Z_n$, where $(Y_n)$ is an $L^1$-bounded martingale and where $(Z_n)$ is a martingale in the limit that goes to zero a.s. in norm. This theorem still holds for a new class that generalizes martingales in the limit. We show that a real-valued $L^1$-bounded pramart $(X_n)$ can be written $X_n = Y_n + Z_n$, where $Y_n$ is an $L^1$-bounded martingale, and $Z_n$ has the following property: For each $\varepsilon > 0$, there is an $m$, an $\Sigma_m$-measurable subset $A$ of $\Omega$, and a supermartingale $(T_n)_{n \geq m}$ on $A$ such that $\int_A T_n dP \leq \varepsilon$ and $|Z_n| \leq T_n$ on $A$ for $n \geq m$.