{} | Tail Dependence of Multivariate Pareto Distributions

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Operator Regular Variation of Multivariate Liouville Distributions

Operator regular variation reveals general power-law distribution tail decay phenomena using operator scaling, that includes multivariate regular variation with scalar scaling as a special case. In this paper, we show that a multivariate Liouville distribution is operator regularly varying if its driving function is univariate regularly varying. Our method focuses on operator regular variation of multivariate densities, which implies, as we also show in this paper, operator regular variation of the multivariate distributions. This general result extends the general closure property of multivariate regular variation established by de Haan and Resnick in 1987

Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation

The high-frequency Helmholtz equation on the entire space is truncated into a bounded domain using the perfectly matched layer (PML) technique and subsequently, discretized by the higher-order finite element method (FEM) and the continuous interior penalty finite element method (CIP-FEM). By formulating an elliptic problem involving a linear combination of a finite number of eigenfunctions related to the PML differential operator, a wave-number-explicit decomposition lemma is proved for the PML problem, which implies that the PML solution can be decomposed into a non-oscillating elliptic part and an oscillating but analytic part. The preasymptotic error estimates in the energy norm for both the $p$-th order CIP-FEM and FEM are proved to be $C_1(kh)^p + C_2k(kh)^{2p} +C_3 E^{\rm PML}$ under the mesh condition that $k^{2p+1}h^{2p}$ is sufficiently small, where $k$ is the wave number, $h$ is the mesh size, and $E^{\rm PML}$ is the PML truncation error which is exponentially small. In particular, the dependences of coefficients $C_j~(j=1,2)$ on the source $f$ are improved. Numerical experiments are presented to validate the theoretical findings, illustrating that the higher-order CIP-FEM can greatly reduce the pollution errors