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

Vasily Melnikov ()
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Vasily Melnikov: Strathcona High School

Journal of Theoretical Probability, 2025, vol. 38, issue 1, 1-25

Abstract: Abstract 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.

Keywords: Convex compactness; Émery topology; $$\sigma $$ σ -localization; 60G44; 60H05; 46A50 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10959-024-01388-4

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