Max-P Aggregation for Max-Stable Processes

• Max-P Aggregation Scheme is a unified framework for representing max-stable processes by replacing the supremum norm with the ℓ^p-norm, establishing a continuum from sum-stable to max-stable regimes.
• It connects the algebraic norm used for aggregation with extremal dependence and noise properties, allowing interpolation between classical de Haan and Reich–Shaby models.
• The scheme provides practical modeling tools with explicit criteria via the stable tail dependence function, influencing aspects such as dependence, ergodicity, and spatial extreme analysis.

The Max-P aggregation scheme refers to a unified family of representations for max-stable processes, based fundamentally on the use of $\ell^p$-norms in the spectral construction of these processes. This framework subsumes both the classical de Haan max-stable process representation (based on the $\ell^\infty$-norm) and the finite-aggregation Reich–Shaby construction, providing a continuous interpolation between the classical sum-stable (logistic, $p\downarrow 1$) and max-stable ($p=\infty$) regimes. Max-P aggregation transparently connects the algebraic norm chosen for aggregation with the extremal dependence structure and noise properties of the resulting process, leading to a broad and tractable toolkit for both theoretical investigations and practical modeling of extremes (Oesting, 2017).

1. $\ell^p$-Norm Representation of Max-Stable Processes

Let $X = \{X(s): s \in S\}$ denote a simple max-stable process with unit Fréchet margins, $P(X(s) \le x) = \exp(-x^{-1})$ for all $x > 0$. Classically, such a process admits the de Haan spectral representation: $$X(s) = \max_{i \in \mathbb{N}} A_i V_i(s) = \|\{A_i V_i(s)\}_{i}\|_\infty$$ where $\{A_i\}$ are points of a Poisson process on $(0, \infty)$ with intensity $a^{-2}da$, and $\{V_i\}$ are i.i.d. copies of a nonnegative process $V$ with $E[V(s)] = 1$.

The Max-P (or $\ell^p$-norm) aggregation scheme generalizes this by replacing $\|\cdot\|_\infty$ with the $\ell^p$-norm and introduces an independent "Fréchet-$p$" multiplicative noise: $$X(s) = \frac{U^{(p)}(s)}{\Gamma(1-p^{-1})} \left(\sum_{i=1}^{\infty}[A_i W_i^{(p)}(s)]^p\right)^{1/p}, \quad s \in S$$ Here, $U^{(p)}(s)$ are iid Fréchet-$p$ variables with $\Phi_p(x) = \exp(-x^{-p})$, $W_i^{(p)}$ are iid copies of a process $W^{(p)}$ with $E[W^{(p)}(s)]=1$, and $\Gamma$ denotes the gamma function (Oesting, 2017).

As $p \rightarrow \infty$, $U^{(\infty)} \equiv 1$ and the classical max-stable spectral representation is recovered. As $p \downarrow 1$, the structure approaches the sum-stable regime, and the finite-dimensional distributions tend toward independence.

2. Dependence Structure and Stable Tail Dependence Function

The stable tail dependence function (STDF), defined for $s_1, \dots, s_n \in S$ by

$$l_{s_1,\dots,s_n}(x_1, \dots, x_n) = -\log P(X(s_i) \le 1/x_i, \, i = 1, \dots, n), \quad x_i \ge 0$$

gives a complete characterization of the dependence structure of $X$.

A central result is that a max-stable process $X$ admits a Max-$p$ ($\ell^p$-norm) representation if and only if the reparameterized function

$$f^{(p)}_{s_1,\dots,s_n}(x_1, \dots, x_n) = l_{s_1,\dots,s_n}(x_1^{1/p}, \dots, x_n^{1/p})$$

is conditionally negative definite. Explicitly, for all $m \ge 1$, any vectors $x^{(1)}, \dots, x^{(m)} \in [0, \infty)^n$ and reals $a_1, \dots, a_m$ with $\sum_{i=1}^m a_i = 0$,

$$\sum_{i, j=1}^m a_i a_j f^{(p)}_{s_1,\dots,s_n}(x^{(i)} + x^{(j)}) \le 0$$

[(Oesting, 2017), Theorem 4.1].

This criterion links the existence of a Max-p representation to spectral properties of the extremal dependence structure.

3. Transformations and Special Constructions

The Max-p family has closure and transformation properties:

• Equivalence across $p$: Any process with an $\ell^p$-representation for some $p \in (1, \infty)$ also admits a de Haan form ($p = \infty$) and can be rewritten for any $q \in (p, \infty]$ with explicit transformations for the noise and spectral processes [(Oesting, 2017), Prop. 3.1].
• Reich–Shaby model: For finite $L$ and deterministic $W^{(p)}$ supported on weights $\{w_l(s)\}$, the construction specializes to the Reich–Shaby max-stable process,

$$X(s) = U^{(p)}(s) \left(\sum_{l=1}^L [B_l w_l(s)]^p \right)^{1/p}$$

with $B_l$ having Laplace transform $E[e^{-tB_l}] = e^{-t^{-1/p}}$ (Oesting, 2017).

• Pure logistic processes: For $W^{(p)}(s)\equiv1$, $X(s)$ is the pure logistic max-stable process, with multivariate logistic finite-dimensional distributions.

4. Interpretation of Parameter $p$ and Limiting Cases

The parameter $p$ governs the trade-off between noise (marginal independence) and extremal dependence:

• $p \downarrow 1$: Max-p aggregation yields asymptotic independence. The limiting finite-dimensional law is the multivariate logistic family, with margins tending to independent unit Fréchet.
• $p = 2$: The bivariate distributions correspond to classical logistic extreme-value distributions with parameter $1/2$. This case connects to "max-Gaussian-noise."
• $p = \infty$: The process reduces to the supremum-based max-stable spectral representation, with no additional noise and maximally strong extremal dependence.

A summary of key structural distinctions:

$p$ Value	Dependence Structure	Representation Form
$p \to 1$	Independence (logistic)	Sum-stable
$p = 2$	Logistic, max-Gaussian	Gaussian kernel interpretation
$p = \infty$	Classical max-stable	de Haan spectral

5. Measures of Dependence, Mixing, and Ergodicity

A process with a Max-p representation satisfies explicit lower bounds on extremal dependence: $$\theta(\{s,t\}) = -\log P(\max\{X(s), X(t)\} \le x) / x \ge 2^{1/p}$$ where $\theta(\cdot)$ is the pairwise extremal coefficient [(Oesting, 2017), Prop. 5.1]. The process denoised of Fréchet-$p$ noise, $\overline{X}(s) = \max_i A_i W_i^{(p)}(s)$, satisfies

$$E[\max\{W^{(p)}(s), W^{(p)}(t)\}] \le \theta(\{s,t\}) \le 2^{1/p} E[\max\{W^{(p)}(s), W^{(p)}(t)\}]^{1-1/p}$$

Mixing and ergodicity on countable index sets $S = \mathbb{Z}$ are determined by the asymptotic/panel-averaged extremal coefficient: $$X \text{ is mixing } \iff \lim_{|h|\rightarrow\infty} \theta(\{0,h\}) = 2$$

$$X \text{ is ergodic } \iff \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^n \theta(\{0,k\}) = 2$$

In the Max-p setting, these properties are determined by the denoised process $\overline{X}$, and the Fréchet-$p$ noise does not affect mixing or ergodicity [(Oesting, 2017), Prop. 6.3].

6. Applications and Model Selection Considerations

The Max-P aggregation scheme is central for constructing flexible max-stable models. The interpolation through $p$ provides direct control over the degree of noise (nugget effect) versus extremal dependence, permitting fine-tuning for modeling spatial and multivariate extremes. The Reich–Shaby construction is commonly used for spatial modeling with a nugget, and the pure logistic case enables simple analytic forms. The conditional negative definiteness of the reparametrized STDF serves as a concrete criterion for model selection and assessment of representability within this framework.

A plausible implication is that Max-P aggregation unifies disparate constructions used across spatial extremes, allowing systematic transitions between them and rigorous characterization of their dependence properties (Oesting, 2017).