Conditions based on conditional moments for max-stable limit laws (Q2271711)

Let \(X_1,X_2,\dots\) be independent and identically distributed random variables with common distribution function (df) \(F\). Let \(M_n=\max\{ X_1,X_2,\dots,X_n\}\) and let \(x_0=\sup\{x:F(x)<1\}\). If there exit a nondegenerate df \(G\) and norming constants \(a_n>0,b_n\) such that the df of a normalized version of \(M_n\) converges to \(G\), i.e. \(\text{Pr}\left((M_n-b_n)/a_n\leq x\right)=F^n(a_nx+b_n)\to G(x), n\to\infty,\) then we say that \(G\) is an extreme value df and \(F\) is in the domain of attraction of \(G\), written as \(F\in D(G)\). The extremal types theorem characterizes the limit df \(G\) as of the type of one of the following three classes: Gumbel \(\Lambda(x)=\exp\{-\exp(-x)\},x\in R\); Fréchet \(\Phi_{\alpha}(x)=\exp\{-x^{-\alpha}\},x\geq0,\alpha>0\); Weibull \(\Psi_{\alpha}(x)=\exp\{-(-x)^{\alpha}\},x<0,\alpha>0\) (see \textit{R. A. Fisher} and \textit{L. H. C. Tippett} [Proc. Camb. Philos. Soc. 24, 180--190 (1928; JFM 54.0560.05)], \textit{B. Gnedenko} [Ann. Math. (2) 44, 423-453 (1943; Zbl 0063.01643)] , \textit{L. de Haan} [On regular variation and Its application to the weak convergence of sample extremes, Amsterdam: Mathematisch Centrum (1970; Zbl 0226.60039); Stat. Neerl. 30, 161--172 (1976; Zbl 0353.60032)], \textit{I. Weissman} [J. Am. Stat. Assoc. 73, 812--815 (1978; Zbl 0397.62034)]). Criteria for \(F\in D(G)\) have been studied through regularly varying functions (see [de Haan, loc. cit. (1970)] and \textit{S. I. Resnick} [Extreme values, regular variation, and point processes, Springer-Verlag (1987; Zbl 0633.60001)]). The necessary and sufficient conditions for \(F\in D(\Lambda)\) based on conditional moments were first established by \textit{A. A. Balkema} and \textit{L. de Haan} [Ann. Probab. 2, 792--804 (1974; Zbl 0295.60014)] and these have been useful in deriving the moment estimator of the tail index (see \textit{A. L. M. Dekkers, J. H. J. Einmahl} and \textit{L. de Haan} [Ann. Stat. 17, No. 4, 1833--1855 (1989; Zbl 0701.62029)]). The result of Balkema and de Haan [loc.cit.] is a particular case of the following due to \textit{J. L. Geluk} [Stat. Probab. Lett. 31, No. 2, 91--95 (1996; Zbl 0879.60055)]: Define \(\mu_p(t)=E\{(X-t)^p| X>t\}\) and \( J_p(t)=\{\Gamma(1+p)\}^{-1}\int_{t}^{x_0}(x-t)^p dF(x)\) for \(t<x_0\). If \(F\in D(\Lambda)\), then \(\mu_p(t)<\infty\) for all \(p\geq0\) and \(\mu_p(t)\mu_{p+2}(t)/(\mu_{p+1}(t))^2\to(p+2)/(p+1)\) as \(t\uparrow x_0\), or equivalently \(J_p(t)J_{p+2}(t)/(J_{p+1}(t))^2\to1.\) Conversely, if \(\mu_p(t)<\infty\) for some \(p\geq0\) and the indicated relation holds true, then \(F\in D(\Lambda)\). In the paper under reviewing the necessary and sufficient conditions for \(F\in D(\Phi_{\alpha})\) and \(F\in D(\Psi_{\alpha})\) in terms of conditional moments are derived. One of the main results is as follows: If \(F\in D(\Phi_{\alpha})\) with \(\alpha>2\), then for all \(0\leq p<\alpha-2,\mu_{p+2}(t)<\infty\) and \(\mu_p(t)\mu_{p+2}(t)/(\mu_{p+1}(t))^2\to(p+2)(\alpha-p-1)/((p+1)(\alpha-p-2))\) as \(t\to\infty\). Conversely, if \(\mu_{p+2}(t)<\infty\) for some \(0\leq p<\alpha-2\) with \(\alpha>2\) and the indicated relation holds true, then \(F\in D(\Phi_{\alpha})\).