Convergence to Fréchet distribution via Mallows distance (Q2006722)

For a sequence of random variables \(X_1,X_2,\ldots\) and a fixed \(\alpha>0\), let \(M_n=n^{-1/\alpha}\max\{X_1,\ldots,X_n\}\) have distribution function \(F_{M_n}\). Let \(Y_1,Y_2,\ldots\) be a sequence of IID Fréchet random variables, each with the distribution function \(\Phi_\alpha(x)=\exp(-x^{-\alpha})\) for \(x>0\). The authors show that if \[ \frac{1}{n}\sum_{j=1}^n\mathbb{E}\left(|X_j-Y_j|^\alpha 1_{\{|X_j-Y_j|>bn^{1/\alpha}\}}\right)\rightarrow0 \] as \(n\rightarrow\infty\) for any \(b>0\), then the Mallows (or Wasserstein) distance \(d_\alpha(F_{M_n},\Phi_\alpha)\rightarrow0\) as \(n\rightarrow\infty\), where \(d_\alpha\) is defined for any distribution functions \(F\) and \(G\) by \[ d_\alpha(F,G)=\inf_{(X,Y)}\left\{\mathbb{E}|X-Y|^\alpha\right\}^{1/\alpha}\,, \] where \(X\) has distribution \(F\), \(Y\) has distribution \(G\), and the infimum is taken over all couplings of \(X\) and \(Y\). This convergence in Mallows distance implies convergence in distribution of \(F_{M_n}\) to \(\Phi_\alpha\). In particular, in the special case where \(X_1,X_2,\ldots\) are IID with common distribution function \(F\), this shows that if \(d_\alpha(F,\Phi_\alpha)<\infty\) then \(d_\alpha(F_{M_n},\Phi_\alpha)\rightarrow0\) as \(n\rightarrow\infty\). Results on convergence of moments of \(M_n\) to those of the Fréchet distribution also follow from this general result on convergence of Mallows distance.