SECOND ORDER REGULAR VARIATION AND ITS APPLICATIONS TO RATES OF CONVERGENCE IN EXTREME-VALUE DISTRIBUTION
The rate of convergence of the distribution of order statistics to the corresponding extreme-value distribution may be characterized by the uniform and total variation metrics. de Haan and Resnick  derived the convergence rate when the second order generalized regularly varying function has second order derivatives. In this paper, based on the properties of the generalized regular variation and the second order generalized variation and characterized by uniform and total variation metrics, the convergence rates of the distribution of the largest order statistic are obtained under weaker conditions.
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