Research On Convex Geometry And Stable Distribution Theory

This thesis mainly studies the relationships between convex geometry theory and multivariate stable distribution theory.In 2008,Ilya Molchanov introduced the relationship between star bodies and multivariate stable distributions,and initially established the relationship between the characteristic function and probability density function of multivariate stable random vectors and its related star body.These studies have a wide range of applications in dealing with finite-dimensional distribution or infinite-dimensional distribution in the process of stable distribution.The main works of this paper are to extend the relationship between the above star bodies and the multivariate stable distribution,verify the new relationship between the distribution of linear combination of independent multivariate stable random vectors and the star body,and give the expression of the bound of probability density function of the multivariate stable random vector.Firstly,under the framework of the theory of multivariate stable distribution,the paper uses the properties of multivariate stable distribution to prove that the linear combination of independent multivariate stable random vectors still obeys the multivariate stable distribution,and proves that the inverse proposition of this proposition is also true.Secondly,in this paper,we extend the relationship between star bodies and multivariate stable distribution given by Ilya Molchanov to a new relationship between star body and the distribution of linear combination of independent multivariate stable random vectors in the n-dimensional Euclidean space,and it is obtained that the distribution of linear combination of multivariate stable random vectors is also uniquely determined by the value of its related star body and its characteristic index.Finally,under the framework of convex geometry theory and uniform distribution theory,this paper creatively gives the expression of the bound of the probability density function of multivariate stable random vector.