Abstract
The following conditions are necessary and jointly sufficient for an arbitrary càdlàg local martingale to be a uniformly integrable martingale: (A) The weak tail of the supremum of its modulus is zero; (B) its jumps at the first-exit times from compact intervals converge to zero in $L^{1}$ on the events that those times are finite; and (C) its almost sure limit is an integrable random variable.
Citation
Hardy Hulley. Johannes Ruf. "Weak tail conditions for local martingales." Ann. Probab. 47 (3) 1811 - 1825, May 2019. https://doi.org/10.1214/18-AOP1302
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