The Asymptotic Expansion Of Tail Dependence Copula At The Origin

This paper mainly based on Archimedean Copulas,discussing the properties of the second-order regular variation and generalized second-order regular variation at the origin.Using the second-order regular variation and the generalized second-order regular variation,the asymptotic expansion of the extreme tail dependence Archimedean copula at the origin is obtained under di(?)erent conditions of the Archimedean generator.The convergence of the extreme tail dependence copula at the origin is theoretically proved,and its convergence is verified by numerical simulation and empirical analysis.The practical usefulness of the results is illustrated in the analysis of the stock market data.We discuss the properties of second-order regular variation and generalized secondorder regular variation at the origin in section 2;In section 3,we proved the asymptotic expansion of extreme tail dependence copula at origin under second-order regular variation,and proved the necessary and su cient conditions for convergence to the independent copula.By applying the generalized second-order regular variation,the asymptotic expansion of the corresponding extreme tail dependence copula is obtained.Simulation studies are presented in section 4.We verify its convergence by constructing the extreme tail dependence copula.Finally,the stock data of Tencent and Nintendo from 2018 to2022 are used to empirically analyze the conclusion.