On tail index estimation using dependent data (Q1178953)

Assume that \(X_ 1,X_ 2,\dots\) is a sequence of dependent random variables with the same marginal distribution function \(F\), where \(1-F\) is regularly varying at \(\infty\), that is, there exists an \(\alpha>0\) such that \[ \{1-F(tx)\}/\{1-F(x)\}\to t^{-\alpha}\hbox { as } x\to\infty\hbox{ for all } t>0. \] The author considers the estimation problem of \(\alpha\) based on \(X_ 1,\dots,X_ n\), where \(-\alpha\) is called the regular variation index of \(1-F\). The Hill estimator \(H_ n\) of \(\alpha^{-1}\) is defined by \[ m^{-1}\sum_{j=1}^ m\log X_{(j)}-\log X_{(m+1)}, \] where, for \(j=1,\dots,n\), \(X_{(j)}\) denotes the \(j\)th largest value of \(X_ 1,\dots,X_ n\). The consistency and asymptotic normality of \(H_ n\) is also obtained. The results are specified to sequences \(\{X_ i\}\) which are strictly stationary and satisfy a certain mixing condition.