Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale
Let for each 𝑛∈ℕ𝑋𝑛 be an ℝ𝑑-valued locally square integrable martingale w.r.t. a filtration (ℱ𝑛(𝑡),𝑡∈ℝ+) (probability spaces may be different for different 𝑛). It is assumed that the discontinuities of 𝑋𝑛 are in a sense asymptotically small as 𝑛→∞ and the relation 𝖤(𝑓(⟨𝑧𝑋𝑛⟩(𝑡))|ℱ𝑛(𝑠))−𝑓(⟨𝑧𝑋𝑛⟩(𝑡))𝖯→0...
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| Format: | Article |
| Language: | English |
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Wiley
2011-01-01
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| Series: | Journal of Probability and Statistics |
| Online Access: | http://dx.doi.org/10.1155/2011/580292 |
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| author | Andriy Yurachkivsky |
| author_facet | Andriy Yurachkivsky |
| author_sort | Andriy Yurachkivsky |
| collection | DOAJ |
| description | Let for each 𝑛∈ℕ𝑋𝑛 be an ℝ𝑑-valued locally square integrable martingale w.r.t. a filtration (ℱ𝑛(𝑡),𝑡∈ℝ+) (probability spaces may be different for different 𝑛). It is assumed that the discontinuities of 𝑋𝑛 are in a sense asymptotically small as 𝑛→∞ and the relation 𝖤(𝑓(⟨𝑧𝑋𝑛⟩(𝑡))|ℱ𝑛(𝑠))−𝑓(⟨𝑧𝑋𝑛⟩(𝑡))𝖯→0 holds for all 𝑡>𝑠>0, row vectors 𝑧, and bounded uniformly continuous functions 𝑓. Under these two principal assumptions and a number of technical ones, it is proved that the 𝑋𝑛's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes (𝑋𝑛(0),⟨𝑋𝑛⟩) converge in distribution to some (∘𝑋,𝐻), then a sequence (𝑋𝑛) converges in distribution to a continuous local martingale 𝑋 with initial value ∘𝑋 and quadratic characteristic 𝐻, whose finite-dimensional distributions are explicitly expressed via those of (∘𝑋,𝐻). |
| format | Article |
| id | doaj-art-77660999ddc242eda5fc33b9294ee2b9 |
| institution | Kabale University |
| issn | 1687-952X 1687-9538 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Probability and Statistics |
| spelling | doaj-art-77660999ddc242eda5fc33b9294ee2b92025-02-03T07:24:52ZengWileyJournal of Probability and Statistics1687-952X1687-95382011-01-01201110.1155/2011/580292580292Convergence of Locally Square Integrable Martingales to a Continuous Local MartingaleAndriy Yurachkivsky0Taras Shevchenko National University, 01601 Kyiv, UkraineLet for each 𝑛∈ℕ𝑋𝑛 be an ℝ𝑑-valued locally square integrable martingale w.r.t. a filtration (ℱ𝑛(𝑡),𝑡∈ℝ+) (probability spaces may be different for different 𝑛). It is assumed that the discontinuities of 𝑋𝑛 are in a sense asymptotically small as 𝑛→∞ and the relation 𝖤(𝑓(⟨𝑧𝑋𝑛⟩(𝑡))|ℱ𝑛(𝑠))−𝑓(⟨𝑧𝑋𝑛⟩(𝑡))𝖯→0 holds for all 𝑡>𝑠>0, row vectors 𝑧, and bounded uniformly continuous functions 𝑓. Under these two principal assumptions and a number of technical ones, it is proved that the 𝑋𝑛's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes (𝑋𝑛(0),⟨𝑋𝑛⟩) converge in distribution to some (∘𝑋,𝐻), then a sequence (𝑋𝑛) converges in distribution to a continuous local martingale 𝑋 with initial value ∘𝑋 and quadratic characteristic 𝐻, whose finite-dimensional distributions are explicitly expressed via those of (∘𝑋,𝐻).http://dx.doi.org/10.1155/2011/580292 |
| spellingShingle | Andriy Yurachkivsky Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale Journal of Probability and Statistics |
| title | Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale |
| title_full | Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale |
| title_fullStr | Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale |
| title_full_unstemmed | Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale |
| title_short | Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale |
| title_sort | convergence of locally square integrable martingales to a continuous local martingale |
| url | http://dx.doi.org/10.1155/2011/580292 |
| work_keys_str_mv | AT andriyyurachkivsky convergenceoflocallysquareintegrablemartingalestoacontinuouslocalmartingale |