Geodesic
On stably weak convergence of semimartingales and point processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 320-332

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Let $\{X_n,\,n=1,2,\dots\}$ be a sequence of random elements defined on the probability space $(\Omega,\mathscr F,\mathbf P)$ and taking values in the separable metric space $\mathfrak X$. Let $\mathscr G$ be a $\sigma$-subalgebra of $\mathscr F$. We find general conditions for the sequence $\{X_n,\,n=1,2,\dots\}$ to converge $\mathscr G$-stably; weakly, i. e. for the sequence $\{\mathbf E[\chi_Af(X_n)],\,n=1,2,\dots\}$ to converge for each $A\in\mathscr G$ and for each continuous bounded function $f$ on $\mathfrak X$. The cases of $\mathscr G$-stably weak convergence of semimartingales and point processes are investigated in detail. 
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     author = {B. I. Grigelionis and R. A. Mikulevi\v{c}ius},
     title = {On stably weak convergence of semimartingales and point processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {320--332},
     publisher = {mathdoc},
     volume = {28},
     number = {2},
     year = {1983},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a6/}
}
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B. I. Grigelionis; R. A. Mikulevičius. On stably weak convergence of semimartingales and point processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 2, pp. 320-332. http://geodesic.mathdoc.fr/item/TVP_1983_28_2_a6/