Glivenko-Cantelli Theorems for integrated functionals of stochastic processes

• Main Authors: LI, Jia, ZHANG, Congshan, LIU, Yunxiao
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2021
• Subjects: Glivenko-Cantelli; high-frequency data; spot volatility; microstructure noise; occupation measure; Econometrics; Probability
• 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
• Institution: Singapore Management University

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.