The Theory Of Cluster Point Process On Gaussian Random Field

With the intensification of human activities,the occurrence of extreme events has increased significantly.Although extreme events are rare,the damage caused by them is serious so that there has been a developing interest in the extreme value theory.At present,the research of extremes can be divided into two categories,one is theoretical research,and the other is practical application research.This thesis mainly focuses on the theory of cluster point processes on Gaussian Random Fields.With the development of science and technology,the amount of data obtained is larger and the data types are becoming more complicated.The research about random fields has become more and more important because random fields can be used to represent various complex data types.It is well known that the Gaussian distribution is common in nature so that the theoretical research on Gaussian random fields is of great practical significance.When the variables are strong dependent,clustering of exceedances may occur.In real life,the cluster phenomenon is also very common,such as the persistent high temperature and rainstorm.When it comes to the cluster phenomenon between random variables,Leadbetter first proposed the cluster point process in 1983.However,the research of extremes for Gaussian random fields mainly focuses on the study of asymptotic distributions and related properties.The research on the cluster point processes is less.This thesis is the first systematic study on the cluster point processes of Gaussian random fields,which is the improvement of the study of extreme value theory and can also provide some new insights for the creation of related space-time models.Based on the existing literature,the asymptotic behavior of the cluster point processes under different conditions are obtained,mainly including the following five situations:one-dimensional stationary Gaussian triangular arrays,two-dimensional stationary Gaussian triangular arrays,multidimensional stationary Gaussian triangular arrays,stationary Gaussian random fields and stationary skew Gaussian random fields on a n × n lattice.In the case of one-dimensional Gaussian triangular arrays,this thesis shows that the cluster point process converges in distribution to a Poisson process under the conditions of Hsing et al.(1996)and the partial sum and the cluster point process are asymptotically independent under certain conditions.Furthermore,a numerical simulation is given to illustrate the results and the asymptotic distribution of the maximum of the one-dimensional stationary Gaussian triangular array is a direct corollary of the results.Next,this thesis considers the asymptotic behaviors of the cluster point processes of two-dimensional Gaussian triangular arrays.First,the limit generating functions of the cluster point processes are obtained.Because of the uniqueness of the generating function,the asymptotic distributions of cluster point processes for one-dimensional Gaussian triangular arrays can be directly obtained.Meanwhile,this thesis derives the partial sums and the cluster point processes of the two-dimensional Gaussian triangular arrays are also asymptotically independent under certain conditions.Furthermore,this thesis extends the results to the multi-dimensional stationary Gaussian triangular arrays and the limit generating function of the cluster point processes are established.It is worth mentioning that Gaussian random fields are one of the very popular tools for the spatiotemporal data modeling and analysis,which can be simply characterized by mean structure and covariance function.Whatmore,there are a large number of researches on parameters and non-covariance functions.The behavior of extremes of Gaussian random fields has many applications,such as galaxy recognition in astronomy,image anomaly discrimination,brain mapping behavior and so on.Based on the research of Gaussian triangular arrays,this thesis proves that the cluster point processes of stationary Gaussian random field on a n × n lattice also asymptotically converges to the Poisson processes in distribution under certain conditions.The result is also extended to the isotropic Gaussian random fields.In order to make the theory more intuitively,a model is constructed and numerical simulations are given to illustrate the theoretical results.In practical applications,it will be found that the standard normal distribution may not be appropriate.Some data types show strong bias and thick tail,such as temperature data,precipitation data,financial stock market data and so on.Therefore,research on skew Gaussian random fields is very necessary.There are many settings for skew Gaussian random fields,such as logarithmic elliptic random fields,T-distributed random fields,and transformed random fields.This thesis is interested in the cluster point processes of stationary skew Gaussian random fields on a n × n lattice,which also includes chi-random fields a n × n lattice.The research for the stationary skew Gaussian random fields is more technical compare with the aforementioned studies.Furthermore,this thesis shows that the distribution of cluster point process converges to the Poisson distribution asymptotically under certain conditions.Meanwhile,the asymptotic distributions of extremes for skew Gaussian random fields can be obtained directly from the theory.This thesis first investigates the asymptotic behaviors of the cluster point processes and the relationship between the partial sums and the cluster point processes for Gaussian random fields under different restrictions.It is a supplement and improvement of the extreme value theory.