Research On Extremes Of Dependent Gaussian Sequences And Processes

As an important part of extreme value theory,the extremes of dependent Gaussian sequences and processes is widely used in environmental sciences,ruin theory,financial econometrics,communication and other fields.This thesis first considers the asymptotics of the maxima and minima of bivariate Gaussian tri-angular arrays.Furthermore,the joint limiting distributions of the normalized maxima and minima over continuous time and uniform grids are investigated for homogeneous Gaussian fields.Finally,the asymptotic behaviors between the point processes of ?-upcrossings and exceedances formed by a centered stationary Gaussian process are established.Details are as follows:Chapter two focus on the asymptotic expansions of the maxima and mini-ma of independent and non-identically bivariate Gaussian triangular arrays.Let{(?ni,?_ni),1?i?n,n?1} be an independent bivariate Gaussian triangular array with E?ni=E?_ni=0,E?ni2=E?_ni2=1.Denote the bivariate maxima Mn componentwise by Mn=?Mn1,Mn2?=(max1?i?n?ni,max1?i?n ?_ni).Simi-larly,the bivariate minima mn is defined componentwise by mn=?mn1,mn2?=(min1?i?n?ni,min1?i?n ?_ni).The joint limiting distributions of Mn and mn are derived as the correlation of(?ni,?_ni)is a function about i/n.Furthermore,second-order expansions of joint distributions of Mn and mn are established.Chapter three considers the joint limiting distribution of the maxima and minima of dependent bivariate Gaussian triangular arrays.Let{?ni,1?i?n,n?1} and {?_ni,1?i?n,n?1} be stationary Gaussian arrays,and their correlations satisfy some conditions.The joint limiting distribution of Mn and mn is derived and the result is similar to Chapter two,which shows that Mn and mn are asymptotically independent.Chapter four investigates the joint limiting distributions of the normalized maxima and minima over continuous time and uniform grids for a centered ho-mogeneous Gaussian random field {X?t?,t?0} with unit variance.Precise-ly,denote MT=?MT,MT??=?maxt?ITX?t?,maxt?IT????i=1dR??i?X?t??and mT=?mT,mT??=)mint?ITX?t?,mint?IT????i=1dR??i?X?t?)be respectively the maxima and minima over continuous time and uniform grids,where d?2,IT=?i=1d[0,Ti]and IT????i=1dR??i?means ?i=1d[0,Ti]???R??i?},with the u-niform grid R??i?={k?i,k?N} is given by?i?2log???Ri?1/?i?Di,i=1,2,…,d as Ti??,i=1,2,…,d.We say that the grid is dense if all Di=0 and if all Di=?,the grid is sparse.The grid is called a Pickands' grid if all Di??0,??.The result shows that MT and mT are asymptotically independent with the choice of a sparse grid,Pickands' grid or dense grid when the correlation function r?t?=Cov?X?t?,X?0??satisfies weakly dependent condition.Chapter five and Chapter six are interested in the asymptotic behaviors be-tween the point processes of ?-upcrossings and exceedances formed by a centered stationary Gaussian process {X?t?,t?0} with unit variance when the corre-lation function r?t?=E?X?0?X?t??satisfies weakly dependent condition and strongly dependent condition,respectively.Let the number of ?-upcrossings be N?,uT?T?=#(?-upcrossings of uT by X?t?,t?T},where T=[0,T],and the number of exceedances be NuT??T?=#{exceedances of uT? by X?t?.t ?T?R???} with R??i?mentioned before.The joint limiting generation functions of N?,uT?T?and NuT??T?are obtained when the grid of the discrete time points is a sparse grid,Pickands' grid or dense grid,respectively.