Asymptotic Distribution Of Maxima And Point Process Of Locally Stationary Gaussian Process

In this paper ,the study is restricted to locally stationary Gaussian process introduced by Berman[1974], which means {X(t),0 < t < T] be a standard Gaussian process , there exists a continuous function c(t), t>0 withand a continuous monotone function K(s) with K(0) = 0,K(s) > 0 (s > 0) such thatFor simplicity ,we assume thatThis implies thatIn this paper ,the asymptotic distribution of the multivariate maxima and point process are investigated.Main results areFor asymptotic distribution of maxima multivariate locally stationary Gaussian process ,we have Theorem 2.2 Let be a multivariate Gaussian process withand if their covariance function and cross-covariancefunction satisfy (2.1)-(2.4), mk(t) satisfy (2.5) and the levels defined as (2.6) thenas T .Define point process NT() such that NT(B) = #{t TB, X{t) = uT, X'(t) > 0}, for arbitrary Borel subset B. The main result isTheorem 3.2 Suppose Gaussian process {X{t),0 < t < T} satisfies (3.1), (3.4),(3.6),(3.7) and the level u satisfies the condition (3.13) ,then the point process Nt() of upcrossing formed by {X(t),0