Operator-Regular Variation in Multivariate Densities

• Operator-regularly-varying density is defined using a positive-definite matrix E and tail exponent p, leading to a quasi-homogeneous limiting behavior under operator scaling.
• This framework models anisotropic tail decay by allowing direction-specific scaling, providing a rigorous approach to detect hidden regular variation in multivariate distributions.
• Applications include multivariate Liouville densities where regularly varying driving functions link operator-based tail behavior with classical heavy-tail models.

Operator-regularly-varying density generalizes classical multivariate regular variation by encoding anisotropic power-law tail behavior through operator scaling rather than scalar scaling. Specifically, a Lebesgue density $f_X:\mathbb{R}^d\to [0,\infty)$ is operator-regularly-varying with respect to a positive-definite index matrix $E$ and tail-exponent $p$ if, after scaling by the operator $t^E=\exp((\log t)E)$, the density converges, under appropriate normalization, to a quasi-homogeneous limit on $\mathbb{R}^d\setminus\{0\}$ as $t\to\infty$. This framework accommodates direction-dependent tail decay rates and extends earlier results on scalar regular variation.

1. Formal Definition

Let $E\in\mathbb{R}^{d\times d}$ be a positive-definite matrix with eigenvalues $\lambda_1,\ldots,\lambda_d>0$ and trace $\operatorname{tr}(E)=\sum_{i=1}^d\lambda_i$. The power-matrix is $t^E = \exp((\log t)E)$, with spectral decomposition $E = O^{-1}DO$, $t^E = O^{-1}t^DO$, where $D=\mathrm{diag}(\lambda_1,\ldots,\lambda_d)$ and $$t^D=\mathrm{diag}(t^{\lambda_1},\ldots,t^{\lambda_d})$$.

A density $f_X$ is operator-regularly-varying with index $E$, tail-exponent $p$, and limit function $\ell$, denoted $f_X\in\mathrm{MRV}(E,-p,\ell)$, if there exists a positive, univariate normalizing function $V(t)\in\mathrm{RV}_p$ and a nonzero $\ell:\mathbb{R}^d\setminus\{0\}\to(0,\infty)$ such that

$$\frac{f_X\bigl(t^E x\bigr)}{t^{-\operatorname{tr}(E)} V(t)} \longrightarrow \ell(x), \quad t\to\infty,$$

locally uniformly for $x\neq0$, and $\ell$ satisfies the quasi-homogeneity property

$$\ell\left(s^E x\right) = s^{-\operatorname{tr}(E)-p}\ell(x),\quad s>0.$$

This generalizes scalar regular variation, the case $E=I$, well-known in heavy-tail theory (Li, 2023).

2. Operator Regular Variation for Multivariate Liouville Densities

A multivariate Liouville distribution with parameters $\alpha_1,\ldots,\alpha_d>0$ and continuous driving function $g:(0,\infty)\to(0,\infty)$, with $\int_0^\infty g(r) r^{d-1}dr<\infty$, has the density

$$f_X(x) = \frac{1}{C} \, g\left(\sum_{i=1}^d x_i\right)\prod_{i=1}^d x_i^{\alpha_i-1}, \qquad x_i>0,$$

where $$C = \int_0^\infty g(r) r^{\sum_{i=1}^d\alpha_i-1}dr>0$$.

If $g$ is univariate regularly varying at infinity with index $-\rho$, i.e., $g\in\mathrm{RV}_{-\rho}$, then for $E=\mathrm{diag}(\alpha_1,\ldots,\alpha_d)$, $\operatorname{tr}(E) = \sum_{i=1}^d \alpha_i$, and normalizing function $V(t)=g(t)t^{\operatorname{tr}(E)}$, one has $V\in\mathrm{RV}_{\operatorname{tr}(E)-\rho}$ and

$$f_X\in\mathrm{MRV}\left(E,-(\rho+\operatorname{tr}(E)),\ell\right),$$

with the limiting density

$$\ell(x) = \frac{1}{C} \left(\sum_{i:\,\alpha_i=\max_j\alpha_j} x_i\right)^{-\rho} \prod_{i=1}^d x_i^{\alpha_i-1}.$$

Thus, anisotropic power-law behavior is encoded by the operator $E$, controlling scaling in each coordinate direction. The occurrence of the largest $\alpha_i$ in the tail normalization underscores directionality in multi-index scaling (Li, 2023).

3. Implications for Tail Decay and Distribution Functions

Operator-regular variation at the density level induces operator-regular variation in the distribution measure. For any Borel set $B\subset\mathbb{R}^d$ bounded away from the origin,

$$\lim_{t\to\infty} \frac{\mathbb{P}\{X\in t^E B\}}{V(t)} = \int_B \ell(x) dx =: \mu(B),$$

where $\mu$ is a homogeneous Radon measure of order $-(\rho+\operatorname{tr}(E))$: $$\mu(t^E B) = t^{-(\rho+\operatorname{tr}(E))}\mu(B).$$ This translates precise operator-scaling at the density level to the probability content of rescaled sets, capturing intricacies of multi-dimensional heavy tails (Li, 2023).

4. Relation to Classical Regular Variation and Closure Properties

For $E=I$, operator-regular variation reduces to the classical de Haan–Resnick closure property for scalar-scaled multivariate densities. The scalability to general positive-definite $E$ permits distinct power-law indices and orientation dependence, generalizing scalar regular variation and illuminating possible “hidden” regular variations not visible via scalar scaling alone. This extension bridges traditional multivariate tail theory and more general operator-based anisotropic frameworks (Li, 2023).

5. Example: Power-law Driver with Slowly Varying Modifier

If the driving function is $g(r)=r^{-\rho}L(r)$ with $L\in\mathrm{RV}_0$, then the normalization function becomes

$$V(t) = t^{\sum_i\alpha_i}g(t^{\alpha^*}) = t^{\sum_i\alpha_i-\rho\alpha^*}L(t^{\alpha^*}),$$

where $\alpha^*=\max_i\alpha_i$, so that $V\in\mathrm{RV}_{\sum_i\alpha_i-\rho\alpha^*}$. The limiting density is

$$\ell(x) = \frac{1}{C}\left( \sum_{i:\alpha_i=\alpha^*} x_i \right)^{-\rho} \prod_{i=1}^d x_i^{\alpha_i-1}.$$

In the isotropic special case $\alpha_1=\cdots=\alpha_d=1$, $E=I$, $\alpha^*=1$, the normalization is $L(t)=t^{d-\rho}L(t)$ and the limiting density simplifies to

$\ell(x) = \frac{1}{\int_0^\infty r^{d-1}g(r)\,dr}.$

This recovers classical multivariate regular variation with tail index $\rho+d$ (Li, 2023).

6. Structural Summary and Distinctive Features

Operator-regularly-varying densities, through operator scaling, unify and extend the reach of regular variation theory to accommodate direction-specific and coordinate-coupled power-law tails. They encompass classical multivariate cases and expose structure missed by scalar-based approaches, yielding precise limiting Radon measures and revealing “hidden” regular variation. The framework is especially apt for models such as multivariate Liouville distributions whose construction admits decomposition along directions with varying index parameters. The proper selection of $E$ and driving functions $g$ determines both the asymptotic decay and the precise directional structure of the tails (Li, 2023).