A sufficient condition for the tightness of a sequence of stochastic processes is given in terms of their behavior after stopping times. As an application, the conditions for McLeish's invariance principle for martingales are weakened.
"Stopping Times and Tightness." Ann. Probab. 6 (2) 335 - 340, April, 1978. https://doi.org/10.1214/aop/1176995579