Answer-Stable Tail (AST) in Heavy-Tailed Models

• AST is a property where the aggregate tail of stable-distributed components remains stable under convolution, even with renewal dependencies.
• It extends classical stable laws by incorporating regenerative structures to explain local clustering and scaling of extreme events.
• The framework introduces explicit extremal indices and tail measures that model phase transitions and cluster-size laws in heavy-tailed time series.

The Answer-Stable Tail (AST) property encapsulates the phenomenon whereby the tail of an aggregate (sum or mixture) of stable-distributed objects is itself stable, even when these aggregates arise from stochastic processes with complex long-range or regenerative dependence. In stable-regenerative multiple-stable models, this property reflects how the addition of independent, heavy-tailed “copies” preserves the stable nature of the extremes’ distribution, while renewal structures govern the detailed clustering and dissipation of extreme events. The AST concept extends classical stable theory by emphasizing both the local shape (via tail processes) and the global frequency (via extremal indices) of extremes, particularly in models built from intersection of renewal processes with infinite mean (Bai et al., 2021).

1. Foundations of Stable Laws and the Tail Property

A random variable $X$ is %%%%1%%%%-stable if its distribution is stable under convolution, parameterized by the tail index $\alpha \in (0,2)$. If $X_1, X_2, \ldots$ are i.i.d. copies, then for $a_n > 0$, $n^{-1/\alpha}(X_1+\dots+X_n)$ converges (in distribution) to a non-degenerate stable law. The “tail property” states that for $x \to \infty$,

$P(|X| > x) \sim L(x)x^{-\alpha}$

where $L(x)$ is slowly varying. In classical settings, sums of such variables inherit this tail behavior—a stability phenomenon fundamental in high-dimensional random projections [0611114].

2. Stable-Regenerative Multiple-Stable Model

The stable-regenerative multiple-stable model rigorously generalizes the AST principle to stationary stochastic processes with intricate dependence. Consider parameters: tail-index $\alpha \in (0,2)$; memory-index $\beta \in (0,1)$ governing a renewal law with infinite mean; and multiplicity $p \in \mathbb{N}$. The process is constructed as follows (Bai et al., 2021):

• Renewal framework: Let $\{\tau_i\}_{i\geq 0}$ be renewal times with $P(D_i > n) \sim C_F n^{-\beta}$, so $E D_i = \infty$. The mass function $u(k) = P(k \in \{\tau_i\}) \sim C k^{\beta-1}$ at large $k$.
• Poisson-point representation: Consider points $(x_i, d_i)$ with intensity $\mu(dx, dd) = \alpha x^{-\alpha-1}\,dx\,\pi(d)$, where $\pi(d)$ is the “delay” measure. Attach to each $(x_i, d_i)$ an independent delayed renewal process.
• Process definition: The process $\{X_k\}_{k\geq 0}$ is defined by

$$X_k = \sum_{1 \leq i_1 < \dots < i_p < \infty} \left(\epsilon_{i_1} \cdots \epsilon_{i_p}\right) (x_{i_1} \cdots x_{i_p}) 1_{\{k \in \tau^{(i_1, d_{i_1})} \cap \cdots \cap \tau^{(i_p, d_{i_p})}\}}$$

This construction merges classical stable aggregation with regenerative structures, resulting in extremes whose local and global clustering is determined by renewal process intersections.

3. Tail Process and Phase Transition in Extremal Behavior

The “spectral” tail process $\Theta = \{\Theta_k\}$ captures the limiting local shape of extremes, conditional on an exceedance at time zero. Formally,

$$\left(\frac{X_k}{|X_0|} \,\Big|\, |X_0| > x\right) \xrightarrow[x\to\infty]{d} (\Theta_k)$$

where $\Theta$ encodes the normalized spatial configuration around an extreme (Bai et al., 2021).

The composite memory exponent $\beta_p = p\beta - (p-1)$ controls a sharp phase transition:

Regime	$\beta_p$ Condition	$\eta$ (Intersection)	$\Theta$ Termination	Extremal Clusters
Supercritical	$\beta_p > 0$	Infinite	Nonterminating	Infinite-length clusters
Critical	$\beta_p = 0$	Infinite (barely)	Nonterminating	Independent-scatter at macro
Subcritical	$\beta_p < 0$	Finite with pos. probability	A.s. terminating	Clusters of finite geometric size

In all regimes, the radial (“Pareto”) tail is preserved, i.e., if $X_0$ has tail index $\alpha$, then cluster maxima and local block maxima inherit this tail—a direct realization of the AST property.

4. Extremal Indices and Clustering of Extremes

Two indices quantify the clustering and regularity of extremes:

• Candidate extremal index (microscopic):

$$\theta_{\mathrm{cand}} = P(\eta_1 = \infty) = \left(\sum_{k=0}^\infty u(k)^p\right)^{-1}$$

Interpretable as the probability no further extremes occur after time zero in a local cluster.

• Actual extremal index (macroscopic):

$$P\left(\max_{k=1,\ldots,n} X_k / b_n \leq x\right) \to \exp(-\theta x^{-\alpha})$$

For $\beta_p < 0$, $\theta = D_{\beta,p} \, q_{F,p}$, where $D_{\beta,p}$ is a signed-difference factor. In particular, for $p=2$, $D_{\beta,2} = 1 - 2\beta$, yielding $\theta_{\mathrm{actual}} < \theta_{\mathrm{cand}}$.

Discrepancy between these indices arises from the global clustering and overlap of extremes rather than local cluster termination alone.

5. Tail Measure and Cluster-Size Laws

The tail measure $\nu$ on the sequence space formalizes both the radial scaling and the cluster-shape determined by $\Theta$:

$$\nu(B) = \int_0^\infty P(r \Theta \in B) \alpha r^{-\alpha-1} dr + \int_0^\infty P(-r \Theta \in B) \alpha r^{-\alpha-1} dr$$

In the subcritical regime, the cluster-length $G \sim \mathrm{Geom}(q_{F,p})$. The point-process law of exceedances in blocks confirms this, with cluster-sizes distributed geometrically, a direct consequence of regenerative construction and the AST property.

6. Implications and Applications of the AST Perspective

The AST property reflects that the tail behavior of sums or aggregates—in these models, aggregates based on complex renewals—remains “answer-stable”: the addition of independent stable-regenerative components yields an aggregate whose extremes are again stable, but whose spatial/temporal clustering is modulated by the parameters $(\alpha, \beta, p)$. The spectral tail process $\Theta$ governs local extremal structure, while extremal indices determine cluster frequency and separation. This elaborates classical stable theory by disentangling the radial (tail) regular variation from the regenerative clustering mechanism.

These insights are foundational for the analysis of extremes in stable random projections [0611114], random sup-measures, and models of heavy-tailed timeseries where regenerative or intersection structures dominate. They provide explicit analytic tools for predicting limit laws, cluster sizes, and tail exponent preservation under complex dependencies.