Multivariate Regularly Varying Copulas

• Multivariate regularly varying copulas are defined by anisotropic tail behavior, where operator tail densities capture varying rates of extremal dependence across dimensions.
• They decompose joint operator regular variation into a dependence component from the copula’s tail density and a marginal component from univariate regular variation.
• The operator framework facilitates explicit modeling with examples like Liouville and Archimedean copulas, aiding analysis in fields such as finance and insurance.

Searching arXiv for the cited papers and closely related work on operator tail dependence and multivariate regularly varying copulas. arxiv_search query: "Operator Tail Densities of Multivariate Copulas (Li, 22 Dec 2025)" Multivariate regularly varying copulas are copulas whose extremal dependence is described by regular variation under possibly anisotropic scaling. In the operator framework, the key object is the operator tail density of the copula, which records how the copula density behaves near a tail corner when different coordinates approach that corner at different regularly varying rates. This yields a decomposition of multivariate operator regular variation into a dependence component carried by the copula and a marginal component carried by univariate regular variation. In this formulation, a multivariate distribution is operator-regularly-varying if and only if, roughly, its copula has an appropriate operator tail density and its marginals are regularly varying with compatible indices (Li, 22 Dec 2025).

1. Conceptual and mathematical setting

Classical multivariate regular variation starts from a nonnegative random vector $X=(X_1,\dots,X_d)$ with distribution $F$. One asks whether there exists a univariate regularly varying function $U$ such that

$\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$

for relatively compact Borel sets $B\subset \mathbb{R}_+^d\setminus\{0\}$ bounded away from $0$. This uses a scalar scaling factor $t$, and is therefore most natural when the marginals are tail-equivalent.

Operator regular variation replaces scalar norming by matrix norming. For a real $d\times d$ matrix $E$, one defines

$t^E=\exp(E\log t).$

The framework developed for multivariate regularly varying copulas restricts to diagonal matrices

$F$0

so that

$F$1

This allows different coordinates to scale at different power rates. For a diagonal operator-regularly-varying mapping $F$2, one has the representation

$F$3

with each $F$4 slowly varying.

At the density level, the central notion is an operator-regularly-varying density. If $F$5 has density $F$6, then

$F$7

means that for some operator-regularly-varying mapping $F$8,

$F$9

locally uniformly on $U$0, where $U$1. The limit $U$2 satisfies the operator homogeneity relation

$U$3

Copulas enter through Sklar’s representation. For continuous marginals $U$4, the copula is

$U$5

and the survival copula $U$6 corresponds to $U$7. Upper-orthant limits of $U$8 near $U$9 correspond to lower-orthant limits of $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$0 near $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$1. This operator scaling viewpoint makes it possible to treat non tail-equivalent marginals without forcing a single radial normalization (Li, 22 Dec 2025).

2. Operator tail densities of copulas

The defining dependence object is the operator tail density of a copula. Let $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$2 have density $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$3. For a tail-order vector $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$4 with $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$5, the upper tail density is the locally uniform limit

$\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$6

where $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$7 and $\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$8. The lower tail density is defined as the upper tail density of the survival copula, equivalently by

$\frac{\mathbb{P}(X\in tB)}{U(t)}\to \mu(B)$9

This definition allows coordinatewise approach to the tail corner at different rates. When all $B\subset \mathbb{R}_+^d\setminus\{0\}$0, it reduces to the classical tail density of Li and Wu (2013). When all $B\subset \mathbb{R}_+^d\setminus\{0\}$1 equal a common value, it reduces to the “tail order” notion in Li and Hua (2015). In operator notation, if $B\subset \mathbb{R}_+^d\setminus\{0\}$2, then the tail density is quasihomogeneous: $B\subset \mathbb{R}_+^d\setminus\{0\}$3

This homogeneity is the copula analogue of the operator homogeneity satisfied by the limiting joint density $B\subset \mathbb{R}_+^d\setminus\{0\}$4. It turns the copula tail density into a margin-free tail descriptor that is adapted to anisotropic scaling rather than scalar scaling. Under continuity of the copula density near the tail corner and local uniform convergence, the limit function is continuous on the interior of $B\subset \mathbb{R}_+^d\setminus\{0\}$5, with extension to the cone closure by continuity. A plausible implication is that operator tail densities play the same structural role for anisotropic extremal dependence that ordinary tail densities play in the scalar case, but now on cones endowed with operator homogeneity (Li, 22 Dec 2025).

3. Decomposition of operator multivariate regular variation

The main structural result is a two-way decomposition linking joint operator regular variation, copula tail densities, and marginal regular variation.

First, if a density is operator-regularly-varying, then the corresponding distribution is operator-regularly-varying in the measure sense. If $B\subset \mathbb{R}_+^d\setminus\{0\}$6, then for Borel sets $B\subset \mathbb{R}_+^d\setminus\{0\}$7,

$B\subset \mathbb{R}_+^d\setminus\{0\}$8

where $B\subset \mathbb{R}_+^d\setminus\{0\}$9 and $0$0. For nonnegative $0$1, this implies that the $0$2-th marginal is regularly varying with tail index $0$3.

Second, operator regular variation of the joint density implies existence of a copula tail density. If $0$4 has ultimately non-increasing density $0$5, copula density $0$6, and

$0$7

then the upper tail density $0$8 exists, and the limiting density of the joint law is recovered from it by

$0$9

where

$t$0

This writes the joint tail density as a transformed copula tail density multiplied by a Jacobian term. The copula contributes the dependence-only part; the Jacobian carries the marginal power-tail exponents.

Third, the reverse implication requires a compatibility condition. Suppose $t$1 has upper tail density $t$2, with $t$3, and suppose the marginals have densities $t$4. If

$t$5

then the joint density is operator-regularly-varying: $t$6 with

$t$7

In this sense, a multivariate regularly varying copula is a copula whose tail density, together with compatible regularly varying marginals, determines an operator-regularly-varying joint law. Conversely, any operator-regularly-varying joint density induces an order-$t$8 copula tail density. If compatibility fails, the paper states that one can still analyze the extremes via copula tail densities and marginal regular variation, but the neat operator-MRV representation fails. This marks compatibility as a structural rather than merely technical condition (Li, 22 Dec 2025).

4. Relation to exponent functions and non-standard regular variation

Operator tail densities are closely related to operator exponent functions. For a copula $t$9 with tail-order vector $d\times d$0, the upper operator exponent function is defined by

$d\times d$1

For copulas with continuous density, the exponent function and the operator tail density are related by

$d\times d$2

where $d\times d$3. Under differentiability,

$d\times d$4

Thus the tail dependence measure generated by the exponent function has density $d\times d$5 on $d\times d$6.

This density–measure relation extends the earlier operator-tail-dependence framework. In that framework, one defines lower and upper operator exponent functions $d\times d$7 and $d\times d$8, and lower and upper operator tail dependence functions $d\times d$9 and $E$0, for a positive-definite matrix $E$1. When $E$2 is diagonal, these functions are homogeneous under $E$3, and if $E$4, the operator construction reduces to standard copula tail dependence with common scaling order (Li, 2016).

The same line of work shows that copulas with operator tail dependence, combined with regularly varying univariate margins, generate a rich class of non-standard multivariate regularly varying distributions. Conversely, under orthant continuity and regularly varying margins, the copula of a non-standard multivariate regularly varying distribution has an upper operator exponent function, and under mild conditions this reduces to standard tail dependence of order 1. This clarifies an important point: anisotropic multivariate regular variation need not imply non-standard copula scaling at the observable copula level. The paper on operator tail densities makes the same point in a density-based form by showing that an operator-regularly-varying joint density induces an order-$E$5 copula tail density after the marginal powers are absorbed into the change of variables. This suggests that much of the anisotropy may be transferred from the copula scale to the marginal scale unless one studies finer cones or hidden regimes (Li, 22 Dec 2025).

5. Representative model classes and explicit constructions

A particularly instructive example is provided by Liouville copulas. A nonnegative random vector $E$6 has a Liouville distribution if, up to normalization,

$E$7

with $E$8 and

$E$9

Although the associated Liouville copula

$t^E=\exp(E\log t).$0

does not have a closed-form expression, it admits an explicit operator tail density. If $t^E=\exp(E\log t).$1, then the Liouville density is operator-regularly-varying, and the induced copula tail density takes the form

$t^E=\exp(E\log t).$2

with

$t^E=\exp(E\log t).$3

In the symmetric case $t^E=\exp(E\log t).$4 and $t^E=\exp(E\log t).$5,

$t^E=\exp(E\log t).$6

The example shows that explicit tail dependence may be available even when the copula itself is analytically intractable (Li, 22 Dec 2025).

A second relevant class is the Laplace-generated multivariate Archimedean copulas. For

$t^E=\exp(E\log t).$7

the extremal regime depends on whether the generator $t^E=\exp(E\log t).$8 is regularly varying, slowly varying, or rapidly varying. If $t^E=\exp(E\log t).$9 with $F$00, then

$F$01

If $F$02, then

$F$03

If $F$04 and satisfies the stated asymptotic condition, then a higher-order tail dependence function exists: $F$05 This classification shows that first-order multivariate regular variation corresponds to the heavy-tailed case, while higher-order tail orders arise for rapidly varying generators (Li, 2024).

A third class is the max-stable / multivariate extreme-value construction based on independent unit Fréchet factors: $F$06 Its copula is

$F$07

This is a max-stable copula, and the associated stable tail dependence function is

$F$08

The model provides a discrete-spectral, max-linear construction of multivariate regularly varying copulas with prescribed pairwise tail dependence coefficients

$F$09

It therefore supplies an explicit constructive mechanism for regularly varying copulas in the extreme-value class (Ferreira, 2012).

6. Assumptions, scope, and open directions

The density-based characterization relies on explicit assumptions. For the direction from operator regular variation of the density to copula tail density, the density is assumed to be ultimately non-increasing. Densities and copula densities are assumed to exist, and continuity near the tail corner is used to guarantee existence and continuity of tail densities. The operator index matrix is restricted to a diagonal matrix with positive entries; the text notes that more general matrices are technically harder because the representation $F$10 may fail. These are substantive constraints on the current theory rather than merely stylistic normalizations (Li, 22 Dec 2025).

The reverse characterization also depends critically on the compatibility condition linking the copula scaling functions $F$11 to the marginal tails $F$12. If the copula and marginals are not compatible, the resulting extremes may not be operator-regularly-varying, even though the copula tail dependence and the marginal regular variation remain analyzable. This separates the existence of a dependence descriptor from the existence of a full operator-MRV representation.

From a modeling viewpoint, the operator framework is designed for situations with non-equivalent marginal tails, such as those encountered in finance and insurance. It shows that once the marginal tail indices $F$13 and the copula operator tail density $F$14 are known, the operator tail density $F$15, the corresponding tail measure, and the joint operator index matrix $F$16 are determined. This suggests a workflow in which marginal tail indices are estimated first, copula tail orders and tail densities are estimated on transformed data, and compatibility is then assessed before reconstructing the joint tail law.

The present theory also points toward hidden multivariate regular variation. The Liouville discussion ends with the conjecture that nontrivial operator tail orders $F$17 for copulas may arise in the analysis of hidden multivariate regular variation on sub-cones of $F$18. A plausible implication is that the order-$F$19 copula tail densities obtained from full-space operator-MRV laws may be only the first layer of a richer hierarchy of anisotropic tail objects, detectable only after restricting to finer cones or hidden regimes (Li, 22 Dec 2025).