Weak Tail Conditions for Local Martingales

The following conditions are necessary and sufficient for an arbitrary c\`adl\`ag local martingale to be a uniformly integrable martingale: (i) The weak tail of the supremum of its modulus is zero; (ii) 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 (iii) its almost sure limit is an integrable random variable.