The Edgeworth expansion for distributions of extreme values
Let \(X_{1}, X_{2}, \cdots \) be a sequence of i.i.d random variables with a common distribution \(F.\) Define \(V=(-\log^{-1}F)^{\leftarrow}\), where \(f^{\leftarrow}(t)=\inf \{ x\in(a,b) : f(x) \geq t \},\) \(\forall t\in (f(a), f(b)).\) The authors consider the rate of convergence problem of extreme values. Under the condition of second order derivative \(V''\) a second order von Mises condition (SOMC) for \(F\) is defined. L. De Haan and S.I. Resnick gave a first order Edgeworth expansion for \(\{ P(\max \{ X_{1}, \dots, X_{n} \} - b_{n})/a_{n} \leq x \}\) by choosing \(a_{n}=nV'(n)\) and \[ b_{n}= \begin{cases} nV'(n) , & \gamma\geq 0;\\ V(\infty)+nV'(n)/\gamma, & \gamma<0,\end{cases} \] and obtained the rate of uniform convergence. If the SOMC is satisfied, then \(V\) has to be a generalized regular variation of second order (GRVSO). In this paper, the authors give a complete solution for the Edgeworth expansion problem when \(V\) does not have any derivative. More precisely, they prove that the GRVSO condition is not only sufficient for the Edgeworth expansion, but also necessary.
scientific article; zbMATH DE number 1782099
scientific article