Geodesic
Necessary conditions for stable convergence of semimartingales
Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 229-232

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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 $t \in\mathbf{R}^+$.
Keywords: semimartingale, independent increments.
Mots-clés : stable convergence 
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     author = {E. Mordecki},
     title = {Necessary conditions for stable convergence of semimartingales},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {229--232},
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     volume = {44},
     number = {1},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a20/}
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E. Mordecki. Necessary conditions for stable convergence of semimartingales. Teoriâ veroâtnostej i ee primeneniâ, Tome 44 (1999) no. 1, pp. 229-232. http://geodesic.mathdoc.fr/item/TVP_1999_44_1_a20/