Glivenko-Cantelli Theorems for integrated functionals of stochastic processes
We prove a Glivenko-Cantelli theorem for integrated functionals of latent continuous-time stochastic processes. Based on a bracketing condition via random brackets, the theorem establishes the uniform convergence of a sequence of empirical occupation measures towards the occupation measure induced b...
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| Main Authors: | , , |
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| Format: | text |
| Language: | English |
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Institutional Knowledge at Singapore Management University
2021
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| Subjects: | |
| Online Access: | https://ink.library.smu.edu.sg/soe_research/2535 https://ink.library.smu.edu.sg/context/soe_research/article/3534/viewcontent/gc_aos.pdf |
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| Institution: | Singapore Management University |
| Language: | English |
| Summary: | We prove a Glivenko-Cantelli theorem for integrated functionals of latent continuous-time stochastic processes. Based on a bracketing condition via random brackets, the theorem establishes the uniform convergence of a sequence of empirical occupation measures towards the occupation measure induced by underlying processes over large classes of test functions, including indicator functions, bounded monotone functions, Lipschitz-in-parameter functions, and Hölder classes as special cases. The general Glivenko-Cantelli theorem is then applied in more concrete high-frequency statistical settings to establish uniform convergence results for general integrated functionals of the volatility of efficient price and local moments of microstructure noise. |
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