On Limiting Distribution Of Partial Sum And Maximum, And Point Process Of Dependent Gaussian Process

The limiting distributions of exceedances on sequences,of upcrossings of high levels related to pro-cesses,and the joint limitiing distribution of sum and maximum for Gaussian process had been considered in this paper respectively.Firstly,let be a standardized stationary Gaussian sequence with , and the point process Nn be the exceedances of random level u'n formed by {Xn,n > 1}. Under some conditions,^ converges in distribution to a Poisson process N on (0, l].At the same time, the limiting distribution of exceedances point process of multi-random levels by {Xn, n > 1} also had been obtained.Secondly, let {X(t), t > 0} be a standardized stationary Gaussian process with bT,and the point process NT*. be the ε- upcrossings of level uT formed by {X(t),t > 0}.When r(t) satisfies some conditions N*T. converges in distribution to a Cox process N on (0, l].Moreover,the limiting distribution of ε-upcrossings point process of more than one level by {X(t),t > 0} had been derived.Finally,let {X(t),t > 0} be a standardized stationary Gaussian process-Suppose that r(t) and(r(t)log t)-1 are both monotone and converge to zero,the joint limiting distribution of St = X(t)dt and M(T) = maxo