This note proves a stronger result than Woodroofe's Theorem (1977) on uniform integrability which is a key tool to obtain second order approximations in certain sequential point and interval estimation problems. The conditions for the stronger conclusion are also weaker than the conditions used by Woodroofe (1977) for his theorem.
"Note on a Theorem of Woodroofe's." Ann. Probab. 9 (6) 1030 - 1033, December, 1981. https://doi.org/10.1214/aop/1176994273