On the domain of attraction of \(\exp (-\exp (-x))\)
Let \(X_1,X_2,\dots\) be independent and identically distributed random variables with distribution function \(F(x)\). Set \(\omega(F)=\sup\{x: F(x)<1\}\), and \(Z_n=\max(X_1,X_2,\dots,X_n)\). We let \(F_t(x)= P(X_1- t\leq x E_t(X_1)\mid X_1>1)\), where \(E_t(X_1)= E(X_1- t\mid X_1>t)\), or more generally, \(\mu_p(t)= E^{(p)}_t(X_1)= E[(X_1- t)^p\mid X_1>t]\). It is a well-established result [see the reviewer, ``The asymptotic theory of extreme order statistics'' (1987; Zbl 0634.62044), Sections 2.1 and 2.4] that, with some linear normalization, \((Z_n- a_n)/b_n\), \(b_n>0\), converges weakly to \(H_0(x)= \exp(-\exp(-x))\) if and only if \(F_t(x)\) converges weakly to the unit exponential distribution. Consequently, one wants to characterize exponentiality among the extreme value distributions (when conditioned) by limiting properties of \(Y_t=(X_1- t)/E_t(X_1)\). The author considers such a characterization by the moments \(\mu_p(t)\). It is well known in such an extreme value setting that exponentiality is characterized by \((*)\) \(\mu_2(t)/\mu^2_1(t)\to 2\). Furthermore, if \(Y_t\) converges weakly to the unit exponential, then \((**)\) \(\mu_p(t)/\mu^p_1(t)\to \Gamma(p+1)\) and thus \((***)\) \(\mu_{p+2}(t)\mu_p(t)/\mu^2_{p+1}(t)\to (p+2)/(p+1)\) (in all limits \(t\to\omega(F)\)). Conversely, if \((***)\) holds for some \(p_0>0\), then by a Helly-Bray argument, combined with uniform integrability, one gets that \((**)\) holds for all integers \(p\) with \(1\leq p\leq p_0\) [see the reviewer, ``Advanced probability theory'' (1995; Zbl 0841.60001), pp. 90-81]. Hence, the general case of \((***)\) reduces to the special case \(p=0\), and exponentiality follows from the cited well-known result. The author uses a more complicated method for proof.
scientific article; zbMATH DE number 1031727