Limit theorems for $σ$-localized Émery convergence
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915052199608320 |
|---|---|
| author | Melnikov, Vasily |
| author_facet | Melnikov, Vasily |
| contents | Given a bounded sequence $\{X^{n}\}_{n}$ of semimartingales on a time interval $[0,T]$, we find a sequence of convex combinations $\{Y^{n}\}_{n}$ and a limiting semimartingale $Y$ such that $\{Y^{n}\}_{n}$ converges to $Y$ in a $σ$-localized modification of the Émery topology. More precisely, $\{Y^{n}\}_{n}$ converges to $Y$ in the Émery topology on an increasing sequence $\{D_{n}\}_{n}$ of predictable sets covering $Ω\times[0,T]$. We also prove some technical variants of this theorem, including a version where the complement of $\{D_{n}\}_{n}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03476 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit theorems for $σ$-localized Émery convergence Melnikov, Vasily Probability Functional Analysis 60G44, 60H05, 46A50 Given a bounded sequence $\{X^{n}\}_{n}$ of semimartingales on a time interval $[0,T]$, we find a sequence of convex combinations $\{Y^{n}\}_{n}$ and a limiting semimartingale $Y$ such that $\{Y^{n}\}_{n}$ converges to $Y$ in a $σ$-localized modification of the Émery topology. More precisely, $\{Y^{n}\}_{n}$ converges to $Y$ in the Émery topology on an increasing sequence $\{D_{n}\}_{n}$ of predictable sets covering $Ω\times[0,T]$. We also prove some technical variants of this theorem, including a version where the complement of $\{D_{n}\}_{n}$ 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. |
| title | Limit theorems for $σ$-localized Émery convergence |
| topic | Probability Functional Analysis 60G44, 60H05, 46A50 |
| url | https://arxiv.org/abs/2408.03476 |