Uniformity And Potter Bounds Of Π-variation Of N-th Order

The equivalent conditions of the domain of attraction of extreme val-ue distribution can be given according to generalized regular variation theory. Using the second order generalized regular variation, the Edge-worth expansion of extreme value distribution is discussed. To get the general Edgeworth expansion of extreme value distribution, more and more attention is given to generalized regular variation of higher order.As a subclass of generalized regular variation of higher order, the behavior of II variation of n-th order is studied in this artical. We derive basic properties of II variation of n-th order based on the generalized regular variation of n-th order. The uniform convergence and existence of potter boundary are proved respectively by contradiction and induc-tion. In addition, A equivalent condition for Π variation of n-th order is given. Based on above results,the selection method of the auxiliary function is given first. The selection method is applied to a special form of Generalized regular variation of n-th order. As a result, the existence of potter boundary and selection method of the auxiliary function are obtained.