On stably weak convergence of semimartingales and point processes

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.