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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sg-smu-ink.soe_research-35342022-03-16T07:35:38Z Glivenko-Cantelli Theorems for integrated functionals of stochastic processes LI, Jia ZHANG, Congshan LIU, Yunxiao 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. 2021-08-01T07:00:00Z text application/pdf https://ink.library.smu.edu.sg/soe_research/2535 info:doi/10.1214/20-AAP1637 https://ink.library.smu.edu.sg/context/soe_research/article/3534/viewcontent/gc_aos.pdf http://creativecommons.org/licenses/by-nc-nd/4.0/ Research Collection School Of Economics eng Institutional Knowledge at Singapore Management University Glivenko-Cantelli high-frequency data spot volatility microstructure noise occupation measure Econometrics Probability |
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Glivenko-Cantelli high-frequency data spot volatility microstructure noise occupation measure Econometrics Probability LI, Jia ZHANG, Congshan LIU, Yunxiao Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
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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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text |
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LI, Jia ZHANG, Congshan LIU, Yunxiao |
author_facet |
LI, Jia ZHANG, Congshan LIU, Yunxiao |
author_sort |
LI, Jia |
title |
Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
title_short |
Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
title_full |
Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
title_fullStr |
Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
title_full_unstemmed |
Glivenko-Cantelli Theorems for integrated functionals of stochastic processes |
title_sort |
glivenko-cantelli theorems for integrated functionals of stochastic processes |
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Institutional Knowledge at Singapore Management University |
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2021 |
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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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