GLIVENKO-CANTELLI THEOREMS FOR INTEGRATED FUNCTIONALS OF STOCHASTIC PROCESSES

• 成果类型: Article
• 署名作者: Lil, Jia; Zhang, Congshan; Liu, Yunxiao
• 署名单位: Duke University; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/20-AAP1637
• 发表日期: 2021
• 页码: 1914-1943
• 关键词: MICROSTRUCTURE NOISE; econometric-analysis; QUADRATIC VARIATION; LIMIT-THEOREMS; volatility; covariation; jumps; time; SEMIMARTINGALES; regression

摘要

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 Holder 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.