On extreme-order statistics and point processes of exceedances in multivariate stationary Gaussian sequences

Joint limit distributions of exceedances point processes and partial sums of gaussian vector sequence

“…As mentioned in Wiśniewski [11], N n are simple point processes. Proof We use the techniques of James et al [16] and Hu et al [19] to prove the result, i.e., to show that there exist point processes N n independent of S n such that…”

“…Define the point process of exceedances N n formed by where B s are Borel sets of (0, 1] (see Wiśniewski [11]). …”

“…Moreover, James et al [16] proved the following result for the asymptotic distribution of normalized random vector (M n , S n ): [16] and Hu et al [19] are used to show that the exceedances point processes and the partial sums are asymptotically independent, from which we can derive the exceedances d-variate point processes defined by Wiśniewski [11] and the joint limiting distributions of the maxima for d-dimensional Gaussian sequences studied by Wiśniewski [10,12].…”

“…Wiśniewski [11] studied the weak convergence of multivariate exceedances point processes formed by multivariate stationary Gaussian sequence. Let {X n , n ≥ 1} be a stationary Gaussian…”

Extremes in Multivariate Mixing Sequences

“…Let {5'" : n G K} denote the sequence of the d-variate point processes of exceedances obtained on the base £ and previously considered normalization (see Wisniewski [11,12] and the one-dimensional case in Leadbetter et al [6]). This means that…”