Limit Distributions Of Order Statistics Under Power Normalization

The content of this thesis is about the limit distributions of order statistics under power normalization, which includes two main parts. The first part mainly focuses on the unified representation of limiting distributions of maxima of indepen-dent and identically distributed random sequence under power normalization, and the limit distributions of order statistics under power normalization and random indice for the second part.It is known that there arc six types power stable laws for the limit distributions of maxima under power normalization as its limit distribution is nongenerate. To establish the unified representation of the power stable laws is interesting in statis-tical theoritical analysis and its applications. The unified representation of the six power stable limit laws is whereτ∈R, the power normalization extreme value index.Meanwhile, some necessary and sufficient conditions, related to general regu-larly varying function or regularly varying function, for the distribution belonging to one type of power stable limit laws are established, which also are useful to jus-tify the distribution belongs to which type of power stable laws as the considered distribution belongs to one type of extreme value distributions.In the second part, we study the limiting distributions of order statistics under power normalization and positive integer random indice as the random index N(n) follows the shifted negative binomial distribution, the shifted Poisson distribution and the shifted binomial distribution, respectively.