Limit Theorems for $$\sigma $$ σ -Localized Émery Convergence

Given a bounded sequence $$\{X^{n}\}_{n}$$ { X n } n of semimartingales on a time interval [0, T], we find a sequence of convex combinations $$\{Y^{n}\}_{n}$$ { Y n } n and a limiting semimartingale Y such that $$\{Y^{n}\}_{n}$$ { Y n } n converges to Y in a $$\sigma $$ σ -localized modification of the Émery topology. More precisely, $$\{Y^{n}\}_{n}$$ { Y n } n converges to Y in the Émery topology on an increasing sequence $$\{D_{n}\}_{n}$$ { D n } n of predictable sets covering $$\Omega \times [0,T]$$ Ω × [ 0 , T ] . We also prove some technical variants of this theorem, including a version where the complement of $$\{D_{n}\}_{n}$$ { D n } n forms a disjoint sequence. Applications include a complete characterization of sequences admitting convex combinations converging in the Émery topology and a supermartingale counterpart of Helly’s selection theorem.