Shift-Generated α-Homogeneous Random Fields

• Shift-generated α-homogeneous random fields are defined by a shift-based invariance property on α-homogeneous observables, ensuring a consistent structure for stationary max-stable fields.
• They employ a jointly measurable setup with normalization constraints (E||Z(0)||^α = 1) and extend to weighted shift identities for integral-type functionals.
• The framework unifies local, spectral, and tail random fields via explicit random-shift constructions and cluster representations, facilitating advanced stationary model design.

Shift-generated (\alpha)-homogeneous random fields are random fields organized through a shift-based functional identity on (\alpha)-homogeneous observables. In the jointly measurable setting, the central object is a class (C[Z]) determined by a fixed exponent (\alpha>0) and a jointly measurable (\mathbb R^d)-valued random field (Z(t)), (t\in \mathbb R^l), on a complete, non-atomic probability space. The significance of these classes is structural rather than merely notational: their elements serve as representors of stationary max-stable random fields, and the same framework links local random fields, spectral tail random fields, tail random fields, and random-shift constructions [2507.18835], [2111.00792], [2206.15064].

1. Formal setup and defining identity

The 2025 formulation fixes (T=\mathbb R^l), a norm (|\cdot|) on (\mathbb R^d), and a jointly measurable field
[
Z(t),\qquad t\in T,
]
subject to
[
\mathbb{P}\Big{\sup_{t\in T}|Z(t)|>0\Big}=1,\qquad \mathbb{E}|Z(0)|^\alpha=1.
]
A notable feature of this setup is that stochastic continuity is not assumed in the general theory; joint measurability is the standing regularity condition [2507.18835].

The class of measurable functionals is
[
\mathcal H=\Bigl{F:(\mathbb R^d)^T\to[0,\infty)\Bigr},
]
measurable with respect to the product (\sigma)-field. For (\beta\ge 0), the subclass (\mathcal H_\beta\subset\mathcal H) consists of the (\beta)-homogeneous maps,
[
F(cf)=c^\beta F(f),\qquad c>0.
]
The shift operator is written
[
B^h f(\cdot)=f(\cdot-h),\qquad h\in T.
]

A random field belongs to a shift-generated (\alpha)-homogeneous class if it satisfies
[
\mathbb{E}{F(Z)}=\mathbb{E}{F(B^h Z)},\qquad \forall F\in\mathcal H_\alpha,\ \forall h\in T.
]
This identity is the defining shift-generated property. It requires invariance of expectations only for (\alpha)-homogeneous observables, not pointwise or finite-dimensional stationarity of (Z) itself. A direct consequence is translation stability: if (Z\in C), then every shift (B^h Z) also belongs to (C) [2507.18835].

The broader homogeneous-class framework introduced earlier uses a general (\alpha)-homogeneous functional (\kappa) in place of the origin-based normalization and defines equivalent classes (\mathscr C_\kappa[Z]) through structural identities and a common tail measure. In that additive-group setting the literature also uses the convention
[
(B^h f)(t)=f(t+h),
]
so the sign in the shift operator is a matter of convention rather than a change in the underlying representation theory [2111.00792].

2. Relation to stationary max-stable random fields

The principal motivation for shift-generated classes is that they generate stationary max-stable random fields. Given independent copies (Z^{(i)}) of (Z) and independent unit exponential variables (\mathcal V_k), the de Haan representation is
[
X_Z(t)=\max_{i\ge 1}\frac{Z^{(i)}(t)}{\left(\sum_{k=1}^i \mathcal{V}_k\right)^{1/\alpha}},\qquad t\in T,
]
with the maximum taken component-wise. The field (Z) is called a representor of (X_Z) [2507.18835].

If (Z\in C), then (X_Z) is stationary, and it has the same law as the max-stable field generated by any shift of (Z). In this sense, the shift-generated identity is the criterion that transfers shift-covariance of (\alpha)-homogeneous functionals on the representor into stationarity of the induced max-stable model. The earlier additive-group theory makes the same point in a tail-measure language: elements of the same shift-invariant homogeneous class are representers of one and the same tail measure, and random shifting preserves the class [2111.00792].

This perspective is especially important because it separates two issues that are often conflated. The representor (Z) need not itself be stationary in the usual finite-dimensional sense; what matters is that it lies in a class for which the (\alpha)-homogeneous functional identity holds. The induced max-stable field then inherits stationarity from the class structure rather than from an a priori stationarity assumption on (Z) [2507.18835].

3. Extension of the shift identity and (L^\alpha)-continuous representatives

A major development in the jointly measurable theory is that the original identity is extended beyond (\mathcal H_\alpha). The paper considers a broader class of maps (F) that can be approximated in probability by finite-dimensional measurable homogeneous functions, and proves equivalences between the original shift identity and weighted shift identities for (0)-homogeneous maps. One key equivalence is
[
\mathbb{E}{Z(h)^\alpha G(Z)}=\mathbb{E}{Z(0)^\alpha G(B^h Z)},\qquad \forall h\in T,\ \forall G\in\mathcal H_0,\ \forall Z\in C,
]
which reformulates the (\alpha)-homogeneous invariance as a weighted shift identity [2507.18835].

The extension is particularly important for integral-type functionals built from
[
\mathcal S_\gamma(f)=\int_T |f(t)|^\alpha \gamma(t)\,\lambda(dt),
]
where (\gamma(t)>0) is a continuous probability density on (T), and (\lambda) is Lebesgue measure. The unweighted notation (\mathcal S) is used when (\gamma\equiv 1). The extended framework includes, for example,

• (F(f)=\Gamma_\beta(f)\mathbf 1{\mathcal S_\gamma(f)=a}\in\mathcal H_\beta),
• (F(f)=\Gamma_\beta(f)\mathcal S_\gamma(f)\in\mathcal H_{\alpha+\beta}),
• (F(f)=\Gamma_\alpha(f)/\mathcal S_\gamma(f)\in\mathcal H_0),

where (\Gamma_\beta\in\mathcal H_\beta). These functionals are central in normalization, random-shift constructions, and integral-statistic arguments [2507.18835].

Another central theorem states that every non-empty shift-generated class contains at least one (L^\alpha)-continuous element:
[
\text{There exists an }L^\alpha\text{-continuous }Z^*\in C,\qquad
\mathbb P{\mathcal S(Z)>0}=1,\ \forall Z\in C.
]
Here (L^\alpha)-continuity means
[
\mathbb E|Z(t_n)-Z(t)|^\alpha\to 0\qquad \text{whenever }t_n\to t.
]
This is weaker than sample-path continuity, but it is sufficient for the structural results of the theory. The proof is constructive in spirit: it uses a representor of a stationary max-stable field, splits the field into positive and negative parts in the one-dimensional case, and extracts an (L^\alpha)-continuous representor [2507.18835].

The same positivity phenomenon appears for local objects:
[
\mathbb P{\mathcal S(\widetilde\Theta)>0}=\mathbb P{\mathcal B(\widetilde Y)>0}=1,
]
where
[
\widetilde Y(t)=R\widetilde\Theta(t),\qquad
\mathcal B(Y)=\int_T \mathbf 1{Y(t)>1}\,\lambda(dt),
]
and (R) is an independent Pareto-type random variable with tail (\mathbb P{R>s}=s^{-\alpha}) for (s\ge 1). This provides the normalization needed for later constructions [2507.18835].

4. Local random fields, spectral tail random fields, and tail random fields

The local version of a shift-generated field is obtained by tilting at the origin. Under
[
\widehat{\mathbb P}(A)=\frac{\mathbb E[Z(0)^\alpha\mathbf 1_A]}{\mathbb E[Z(0)^\alpha]},\qquad A\in\mathscr F,
]
the field
[
\widetilde\Theta=\frac{Z}{Z(0)}
]
is considered. This local field captures the shape of (Z) relative to the origin, and all local random fields corresponding to a fixed (C) have the same law [2507.18835].

The spectral tail random field (\Theta) is characterized by three properties:

1. (\mathbb P{\Theta(0)=1}=1),
2. (\mathbb P{\mathcal S(\Theta)>0}=1),
3. for all (\Gamma\in\mathcal H_0) and (h\in T), [ \mathbb E\bigl{\Theta(h)^\alpha \Gamma(\Theta)\bigr} = \mathbb E\bigl{\mathbf 1{\Theta(-h)\neq 0}\Gamma(B^h\Theta)\bigr}. ]

This is the local form of the shift identity. The theory also proves a converse: any spectral random field is the local field of some shift-generated class (C), and there exists an (L^\alpha)-continuous spectral random field with the same finite-dimensional distributions [2507.18835].

The tail random field is then
[
Y(t)=R\Theta(t),
]
where (R) is independent and Pareto-type. A fundamental identity is
[

\mathbb E\left{\Gamma(xB^hY)\mathbf 1{xY(-h)>1}\right}

x^\alpha \mathbb E\left{\Gamma(\widetilde Y)\mathbf 1{\widetilde Y(h)>x}\right},
\qquad \forall \Gamma\in\mathcal H,\ \forall h\in T,\ \forall x>0.
]
This identity links shifted tail fields and local tail fields and is central in the theory of regularly varying stationary random fields [2507.18835].

The 2021 homogeneous-class framework formulates the same circle of ideas with a general (\kappa)-normalization. In that setting the local field (\Theta) satisfies a generalized time-change formula, while (Y=R\Theta) is characterized by a scaling identity and the condition that (R=\kappa(Y)^{1/\alpha}) is Pareto. The two formulations agree in the special case (\kappa(f)=f(0)^\alpha), which recovers the classical spectral tail process normalization [2111.00792].

Object	Definition	Role
(Z)	jointly measurable representor	generates the class (C[Z])
(\widetilde\Theta) or (\Theta)	(Z/Z(0)) under the tilted law	local or spectral shape at the origin
(Y)	(R\Theta)	tail random field
(Q)	cluster random field	random-shift generator of the same class

5. Random-shift constructions and cluster random fields

The jointly measurable theory uses (\mathcal S_\gamma) and a random shift (N) with density (\gamma) to construct new elements of (C). When (\mathcal S(Z)>0) almost surely, the paper defines a random-shift representor (Z_N) and proves that (Z_N\in C); it also introduces a local-field version (Z_N') and a further variant (Z_N'') based on local cluster structure when (Z(0)) may vanish. These constructions show that the class is not only closed under deterministic shifts but also rich in explicit normalized random-shift representors [2507.18835].

The 2022 cluster-field framework makes this construction systematic. A random field (Q) is a cluster random field if
[
\mathbb P\Bigl{\sup_{t\in T}Q(t)>0\Bigr}=1,
]
and
[
\int_T \mathbb E\Bigl[\sup_{t\in[-a,a]^l\cap T}Q(t-v)^\alpha\Bigr]\lambda(dv)<\infty,\qquad \forall a>0.
]
Given an independent (T)-valued random shift (N) with positive density (p_N(t)>0), the random-shift transform is
[
Z_N(t)=\frac{B^NQ(t)}{[p_N(N)]^{1/\alpha}},\qquad t\in T.
]
The class generated by such transforms is shift-invariant; conversely, if (Q) is a cluster random field, the resulting class is shift-invariant and purely dissipative, and every element has finite positive total (\alpha)-mass,
[
\mathbb P{S\in(0,\infty)}=1,\qquad S:=\int_T Q(t)^\alpha\,\lambda(dt).
]
The class does not depend on the particular choice of the positive-density shift variable (N) [2206.15064].

Cluster random fields unify the original representor (Z), the spectral tail field (\Theta), and the tail field (Y) within a single random-shift representation. The 2022 paper gives explicit constructions of (Q) from each of these objects, including anchored and shift-involution-based versions. It also proves that when (\mathbb P{S<\infty}=1), the induced tail measure admits a cluster/random-shift form and the associated stationary max-stable field has a Rosiński / mixed moving maxima representation
[
X(t)=\max_{i\ge 1}P_i\,Q^{(i)}(t-\tau_i).
]
Thus the finite total cluster mass condition is equivalent to purely dissipative behavior and to the existence of a random-shift representation [2206.15064].

6. Broader homogeneous-class theory and related models

The 2021 theory places shift-generated classes inside a more general family of (\alpha)-homogeneous equivalent classes determined by a measurable map
[
\kappa:D\to[0,\infty),\qquad \kappa(cf)=c^\alpha \kappa(f),\ c>0.
]
For a random field (Z), the class (C=\mathscr C_\kappa[Z]) consists of all representers with the same structural identities and the same tail measure
[
\nu_Z[F]=\int_0^\infty \mathbb E[F(zZ)]\,\alpha z^{-\alpha-1}\,dz.
]
Equality of classes is therefore equality of tail measures, not merely similarity of path properties. When (T) is an additive group, a class is shift-invariant if
[
B^hV\in C,\qquad \forall h\in T,
]
and then random shifts preserve the class:
[
Z_N:=B^N Z\in C
]
for every independent (T)-valued random variable (N) [2111.00792].

This general framework introduces universal maps (U), including
[
\mathcal S_V(f)=\int_V \kappa(B^{-t}f)\,\lambda(dt)
]
and
[
\mathcal B_{V,\tau}(f)=\int_V \kappa(B^{-t}f)^\tau\,\mathbf 1_{{\kappa(B^{-t}f)>1}}\,\lambda(dt),
]
which are used to study dissipativity, conservativity, maximal indices, and extremal asymptotics. For max-stable fields, when (Z) has nonnegative components, the class (C) is shift-invariant exactly when the associated max-stable field is stationary. The same theory covers Brown–Resnick classes, Brown–Resnick–Lévy classes, random-shifted density-based constructions, and symmetric (\alpha)-stable random fields [2111.00792].

Within this development, the 2025 jointly measurable theory represents an extension from the stochastically continuous setting to a broader regularity class. Its main contributions are the shift-generated identity for jointly measurable fields, the extension of that identity to integral and other nontrivial functionals, the existence of an (L^\alpha)-continuous representative in every non-empty class, the converse theory for local and spectral tail fields, and explicit random-shift constructions. A plausible implication is that the framework makes representation theory for stationary max-stable and regularly varying random fields less dependent on sample-path regularity and more directly tied to measurable shift identities and tail structure [2507.18835].