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{} | 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 -th order CIP-FEM and FEM are proved to be under the mesh condition that
is sufficiently small, where is the wave number, is the mesh size, and
is the PML truncation error which is exponentially small. In
particular, the dependences of coefficients on the source are
improved. Numerical experiments are presented to validate the theoretical
findings, illustrating that the higher-order CIP-FEM can greatly reduce the
pollution errors
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