Extremes Of Gaussian Random Fields Approximated By Berman's Method

Extreme Value Theory is of great significance in real life,and the Extreme Value Theory based on Gaussian processes and Gaussian random fields,which is widely used in social science,natural disasters,random impact in economic and financial fields,engi-neering research,meteorological analysis and many other fields closely related to human life,plays an indispensable role in the theory.In this thesis,the main work is to explor the limit behavior in the case of Gaussian random fields by using a very novel method that was innovated by Simeon M Berman who has made an important contribution to the study of extrema of random processes.Berman has used this method to study the tail asymptotics of extrema for centered stationary Gaussian processes,and extended the method to many other processes.As far as I know,the relevant historical literature has not used this method to deeply explore the limit behavior in the case of Gaussian random field.Therefore,this thesis is devoted to explore the extrema of Gaussian random fields by extending Berman's approach.Assume that {X(t),t?[0,T]d} is a stationary Gaussian random field with con-tinuous trajectories,zero mean,unit variance and correlation function r(t)=r(|t|),where|t|=(|t1|,|t2|,...,|td|).Define the sojourn time in the case of Gaussian random field as Lu(T)=?t?[0,T]d I(X(t)>u)dt,u>0,where I(·)is an indicator function.Firstly,we abtained the tail asymptotic of the scaled sojourn time ?(u)Lu(T)as u??,where ?(u)is a scaling function determined by corre-lation function r(t).Then,based on the relationship between sojourn time and extrema of Gaussian random field we finally got the tail asymptotic of the extrema of Gaussian random field over[0,T]d,where BW=?0+ ?y-1 dGh(y)is the Berman's constant and h(t),t?Rd is a continuous function determined by correlation function r(t).W(t),t?Rd is a zero mean Gaussian random field satisfying Var(W(t)-W(s))=2h(t-s).Compared with the Berman's method,this thesis gives a more concise assumption for r(t).At the same time,Sudakov-Fernique inequality,Borell-TIS inequality and a new form of Gaussian weak convergence proof method are used during the proof,which effec-tively simplified the whole proof process and enriched Berman's original proof method.