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Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series
Author(s) -
Jirak Moritz
Publication year - 2016
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/jtsa.12189
Subject(s) - quantile , mathematics , series (stratigraphy) , rate of convergence , empirical distribution function , autoregressive conditional heteroskedasticity , econometrics , context (archaeology) , weak convergence , convergence (economics) , distribution (mathematics) , statistics , mathematical analysis , volatility (finance) , economics , computer science , paleontology , computer network , channel (broadcasting) , computer security , asset (computer security) , biology , economic growth
Given a stationary sequenceX kk ∈ Z , we are interested in the rate of convergence in the central limit theorem of the empirical quantiles and the empirical distribution function. Under a general notion of weak dependence, we show a Berry–Esseen result with optimal rate n −1/2 . The setup includes many prominent time series models, such as functions of ARMA or (augmented) GARCH processes. In this context, optimal Berry–Esseen rates for empirical quantiles appear to be novel.