ANOTHER CLASS OF CONVERGENCE THEOREMS OF MARTINGALELIKE SEQUENCES

Let $\left(x_n, \mathscr{F}_n\right)$ be an integrable adapted sequence of $\mathrm{r} . \mathrm{v}$.'s, and $\bar{T}, T$ the sets of all stopping times and bounded stopping times, respectively. In this paper we prove that (1) if $\left(x_n, \mathscr{F}_n\right)$ is a subpramart which satisfies the assumption $\left(d_T^{+}\right): \frac{\lim }{F} E x_\tau^{+}< \infty$ or $\left(O^{+}\right)$: $\int_{(\tau-\infty)} x_\tau^{+}< \infty, \forall \tau \in \bar{T}$, then $\left(x_n\right)$ converges a.e.; (2) if ( $x_n, \mathscr{F}_n$ ) satisfies the assumption $\left(C^{+}\right)$or $\left(d^{+}\right): \frac{\lim }{n} E x_n^{+}< \infty$ and $\left[E\left(x_\tau \mid \mathscr{F}_n\right)-x_n\right]^{-} \xrightarrow{P} 0$ (pr. ), then ( $x_n$ ) lower demiconverges a. $\boldsymbol{\theta}$..