Necessary conditions for stable convergence of semimartingales

We prove an inverse to a theorem on stable convergence of semimartingales due to Feigin [Stochastic Process. Appl., 19 (1985), pp. 125–134]. As a consequence, it can be stated (under some control in the jumps) that a sequence of martingales $X^n$ converges stably to a continuous martingale $X$ with conditionally independent increments if and onlyif the quadratic variations of $X^n$ converge in probability to the quadratic variation of $X$ for each $t \in\mathbf{R}^+$.