Conditional Extreme Value Theory

• Conditional Extreme Value Theory is a framework that models the joint behavior of extremes by conditioning on one variable, revealing hidden dependence structures.
• It extends classical multivariate extreme value theory by using specialized normalization techniques to capture residual dependence under asymptotic independence.
• It provides explicit asymptotic results for functionals like the product of variables across different tail regimes, aiding in risk assessment and complex system analysis.

Conditional extreme value theory addresses the behavior of stochastic systems or random vectors under the event that one or several components are extreme, providing refined asymptotic results that improve upon or complement classical multivariate extreme value theory (MEVT) especially under asymptotic independence. It explores both the probabilistic structure and inference for conditional tails, dependence, and functionals, often under complex scenarios including varying domains of attraction (Fréchet, Weibull), underlying dependence structures, or physical constraints.

1. Foundations of the Conditional Extreme Value Model (CEVM)

The core of CEVM is to characterize the joint distribution of a random vector, e.g., $(X, Y)$, after appropriate normalization and conditioning on one component (say, $Y$) being extreme. The model assumes:

• The marginal $Y$ is in the domain of attraction of a generalized extreme value distribution $G_\gamma$, so for normalizing functions $a(t)$ (scale) and $b(t)$ (location), $t P[Y > a(t) y + b(t)] \to -\log G_\gamma(y)$ for $y$ in the appropriate set $E(\gamma)$.
• There exist centering and scaling functions $B(t)$ and $\alpha(t)$ such that

$$t P\left( \frac{X - B(t)}{\alpha(t)}, \frac{Y - b(t)}{a(t)} \in \cdot \right) \to \mu(\cdot)$$

with $\mu$ a non-null Radon measure (typically on $[–\infty, \infty] \times E(\gamma)$). This convergence is vague and can occur on cones appropriate to the normalization regime.

• The conditional distribution of $X$ (given $Y$ is extreme) is non-degenerate: for each $y$, the function $x \mapsto \mu((–\infty, x] \times (y, \infty))$ is non-degenerate.

This leads to the notation that $(X, Y) \in \text{CEVM}(\alpha, B; a, b; \mu)$ and implies that after conditioning and suitable normalization, the limiting measure $\mu$ often retains subtle dependencies that are not seen in standard MEVT, particularly in regimes of asymptotic independence.

2. Contrasts with Classical Multivariate Extreme Value Theory

The operational distinction is as follows:

Framework	Conditioning	Marginal assumptions	Limiting measure
Classical MEVT	Focus on joint maxima	All marginals in univariate domain of attraction	Max-stable, homogeneous of order $-1$
CEVM	Conditional on one variable being extreme	Only the conditioning variable required to be in a domain of attraction	General Radon measure $\mu$, not necessarily a product

In classical MEVT, the scaling is coordinated across all components, so only simultaneously large values are considered, and independence in the limit (i.e., the limit is a product measure) signals a loss of dependence structure in extremes. In contrast, the CEVM framework captures “residual” dependence by exploring the conditional tail of the non-conditioning components, revealing dependencies that persist even under asymptotic independence in the classical sense.

3. Tail Behavior of Products in the Conditional Extreme Value Setting

A central technical advancement in (Hazra et al., 2011) is the detailed analysis of the product $XY$ when $(X, Y)$ is in CEVM. The precise asymptotics hinge on the tail indices $p$ (for $X$) and $\gamma$ (for $Y$):

• $p > 0$, $\gamma > 0$ (both heavy-tailed, Fréchet domains):
  • $XY$ exhibits regularly varying tails with index $-1/(p+\gamma)$.
  • $P[XY > u] \sim L(u) u^{-1/(p+\gamma)},$
  • where $L(u)$ is slowly varying.
• $p$ or $\gamma$ negative (Weibull domains):
  • Analysis must consider how close $X$ or $Y$ is to its upper endpoint. For example, with both having finite right endpoints (after appropriate transformation to $(0,1)$), Theorem 5.2 shows $(1-XY)^{-1}$ or $(XY)^{-1}$ is regularly varying, and the tail index—$-1/|p|$ or $-1/(|p|+|\gamma|)$—follows according to the configuration of the endpoints and tail weights.

The general structure is that, after normalization (scaling by $\alpha(t), a(t)$ etc.), one studies the behavior over sets like $$A_{\varepsilon,z} = \{ (x, y): xy > z, y > \varepsilon \}$$ and connects the asymptotic to the limit measure $\mu$ via

$$t P\left( \frac{X}{\alpha(t)}, \frac{Y}{a(t)} \in A_{\varepsilon,z} \right) \to \mu(A_{\varepsilon,z}).$$

A spectral integral representation is available when both marginals can be made nonnegative, so

$$\lim_t t P\left[ \frac{XY}{\alpha(t)a(t)} > z \right] = \int_{[0,1)} \Phi(z; w) S(dw),$$

where $S$ is a spectral measure and $\Phi$ is a kernel encoding the product dependence.

4. Normalization and Domains of Attraction: Fréchet versus Weibull

The selection of normalization functions $a(t)$ and $\alpha(t)$ is dictated by the marginal domains of attraction:

• $Y$ Fréchet ($\gamma > 0$):
  • $P[Y > y] \in RV_{-1/\gamma}$; typically $a(t)\in RV_\gamma$.
• $Y$ Weibull ($\gamma < 0$):
  • $Y$ has finite right endpoint $b(0)$; $a(t)$ is chosen to scale distances from this endpoint: $a(t) = b(0) - b(t)$.

Analogous statements apply to $X$ with its tail index $p$. These normalization conventions ensure that the joint conditional tail converges to a non-degenerate $\mu$, and the tail of $XY$ or its transformation is determined by the combination of $p$, $\gamma$, and the structure of the endpoints, often resulting in the tail index being $-1/(p+\gamma)$ in dual-Fréchet, or $-1/(|p|+|\gamma|)$ in dual-Weibull, or governed by the heavier tail otherwise.

5. Non-Product Limit Measures and Residual Dependence

A pivotal insight is that, in the CEVM, the limit measure $\mu$ is typically non-product, preserving residual dependence even when classical MEVT would predict asymptotic independence. Measure-theoretic methods, specifically vague convergence on cones, are used throughout to demonstrate this property and rigorously justify conditioning arguments.

Practical computation of the tail index and limiting measure for the product is frequently entirely explicit, and, as the examples in Section 7 of (Hazra et al., 2011) show, can provide tail probabilities, even in situations where standard moment conditions fail.

6. Classification of Product Tail Indices

A classification can be organized to summarize the variety of tail behaviors:

$p$	$\gamma$	Transformation	Tail index
$> 0$	$> 0$	$XY$	$-1/(p+\gamma)$
$< 0$	$< 0$	$(XY)^{-1}$	$-1/(|p|+|\gamma|)$
$> 0$	$< 0$	see Theorem 5.6	governed by variable/centering
$< 0$	$> 0$	see Theorem 5.7	governed by heavier tail variable

All these results collectively allow for explicit asymptotic quantification such as

$P[XY > u] \sim L(u) u^{-1/(p+\gamma)}$

in the standard dual-Fréchet case, while other scenarios require analyzing inverses or other transformations according to the support and tail indices.

7. Implications and Applications

The characterization of products in CEVM is important in settings where the simple product form of the limit measure in classical MEVT fails to capture meaningful dependence, particularly under asymptotic independence. The explicit computation of tail probabilities and indices has direct relevance in areas such as risk management in finance and insurance, modeling of network traffic, and any domain where products of heavy-tailed (or bounded upper endpoint) risks must be assessed under conditioning on extreme events.

The rigorous measure-theoretic formulation and the classification of normalization regimes represent a systematic approach that can be generalized to higher dimensions and more complex dependence patterns within the CEVM framework. The methods of (Hazra et al., 2011) thus extend the practical and theoretical reach of conditional multivariate extremes beyond the limitations of classical approaches.