Necessary Conditions for Stable Convergence\\ of Semimartingales

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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 $t \in$ {\bf R}$^+$.