Joint FCLT for Sample Quantile and Measures of Dispersion for\n Functionals of Mixing Processes

2021/11/15 by Marcel Bräutigam, Bräutigam, Marcel, Marie Kratz +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 #60F17 #60G10 #62H10 #62H20 #Advanced Statistical Process Monitoring #Applied mathematics #Autoregressive conditional heteroskedasticity #Bivariate analysis #Central limit theorem #Dispersion (optics) #Econometrics #FOS: Mathematics #Financial Risk and Volatility Modeling #Independent and identically distributed random variables #Joint (building) #Limit (mathematics) #Mathematical analysis #Mathematics #Mixing (physics) #Moment (physics) #Physics #Quantile #Random variable #Sample (material) #Series (stratigraphy) #Statistical Methods and Inference #Statistical physics #Statistics #Statistics Theory (math.ST) #Volatility (finance)

paper · pdf · doi:10.48550/arxiv.2111.07650

openalex publication_date 2021/11/15 · openalex created_date 2022/07/25 · openalex updated_date 2026/08/05

Abstract

In this paper, we establish a joint (bivariate) functional central limit\ntheorem of the sample quantile and the r-th absolute centred sample moment\nfor functionals of mixing processes. More precisely, we consider L₂-near\nepoch dependent processes that are functionals of either \φ-mixing or\nabsolutely regular processes. The general results we obtain can be used for two\nclasses of popular and important processes in applications: The class of\naugmented GARCH(p,q) processes with independent and identically distributed\ninnovations (including many GARCH variations used in practice) and the class of\nARMA(p,q) processes with mixing innovations (including, e.g., ARMA-GARCH\nprocesses). For selected examples, we provide exact conditions on the moments\nand parameters of the process for the joint asymptotics to hold.\n

Citations

• Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation
• Some Limit Theorems for Stationary Processes
• Conditional Heteroskedasticity in Asset Returns: A New Approach
• Generalized autoregressive conditional heteroskedasticity
• A long memory property of stock market returns and a new model
• Establishing conditions for the functional central limit theorem in nonlinear and semiparametric time series processes

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