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 (Hashorva, 24 Jul 2025, Hashorva, 2021, Hashorva, 2022).

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

$\alpha$0

subject to

$\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 (Hashorva, 24 Jul 2025).

The class of measurable functionals is

$\alpha$2

measurable with respect to the product $\alpha$3-field. For $\alpha$4, the subclass $\alpha$5 consists of the $\alpha$6-homogeneous maps,

$\alpha$7

The shift operator is written

$\alpha$8

A random field belongs to a shift-generated $\alpha$9-homogeneous class if it satisfies

$C[Z]$0

This identity is the defining shift-generated property. It requires invariance of expectations only for $C[Z]$1-homogeneous observables, not pointwise or finite-dimensional stationarity of $C[Z]$2 itself. A direct consequence is translation stability: if $C[Z]$3, then every shift $C[Z]$4 also belongs to $C[Z]$5 (Hashorva, 24 Jul 2025).

The broader homogeneous-class framework introduced earlier uses a general $C[Z]$6-homogeneous functional $C[Z]$7 in place of the origin-based normalization and defines equivalent classes $C[Z]$8 through structural identities and a common tail measure. In that additive-group setting the literature also uses the convention

$C[Z]$9

so the sign in the shift operator is a matter of convention rather than a change in the underlying representation theory (Hashorva, 2021).

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 $\alpha>0$0 of $\alpha>0$1 and independent unit exponential variables $\alpha>0$2, the de Haan representation is

$\alpha>0$3

with the maximum taken component-wise. The field $\alpha>0$4 is called a representor of $\alpha>0$5 (Hashorva, 24 Jul 2025).

If $\alpha>0$6, then $\alpha>0$7 is stationary, and it has the same law as the max-stable field generated by any shift of $\alpha>0$8. In this sense, the shift-generated identity is the criterion that transfers shift-covariance of $\alpha>0$9-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 (Hashorva, 2021).

This perspective is especially important because it separates two issues that are often conflated. The representor $\mathbb R^d$0 need not itself be stationary in the usual finite-dimensional sense; what matters is that it lies in a class for which the $\mathbb R^d$1-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 $\mathbb R^d$2 (Hashorva, 24 Jul 2025).

3. Extension of the shift identity and $\mathbb R^d$3-continuous representatives

A major development in the jointly measurable theory is that the original identity is extended beyond $\mathbb R^d$4. The paper considers a broader class of maps $\mathbb R^d$5 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 $\mathbb R^d$6-homogeneous maps. One key equivalence is

$\mathbb R^d$7

which reformulates the $\mathbb R^d$8-homogeneous invariance as a weighted shift identity (Hashorva, 24 Jul 2025).

The extension is particularly important for integral-type functionals built from

$\mathbb R^d$9

where $Z(t)$0 is a continuous probability density on $Z(t)$1, and $Z(t)$2 is Lebesgue measure. The unweighted notation $Z(t)$3 is used when $Z(t)$4. The extended framework includes, for example,

• $Z(t)$5,
• $Z(t)$6,
• $Z(t)$7,

where $Z(t)$8. These functionals are central in normalization, random-shift constructions, and integral-statistic arguments (Hashorva, 24 Jul 2025).

Another central theorem states that every non-empty shift-generated class contains at least one $Z(t)$9-continuous element: $t\in \mathbb R^l$0 Here $t\in \mathbb R^l$1-continuity means

$t\in \mathbb R^l$2

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 $t\in \mathbb R^l$3-continuous representor (Hashorva, 24 Jul 2025).

The same positivity phenomenon appears for local objects: $t\in \mathbb R^l$4 where

$t\in \mathbb R^l$5

and $t\in \mathbb R^l$6 is an independent Pareto-type random variable with tail $t\in \mathbb R^l$7 for $t\in \mathbb R^l$8. This provides the normalization needed for later constructions (Hashorva, 24 Jul 2025).

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

$t\in \mathbb R^l$9

the field

$T=\mathbb R^l$0

is considered. This local field captures the shape of $T=\mathbb R^l$1 relative to the origin, and all local random fields corresponding to a fixed $T=\mathbb R^l$2 have the same law (Hashorva, 24 Jul 2025).

The spectral tail random field $T=\mathbb R^l$3 is characterized by three properties:

1. $T=\mathbb R^l$4,
2. $T=\mathbb R^l$5,
3. for all $T=\mathbb R^l$6 and $T=\mathbb R^l$7,

$T=\mathbb R^l$8

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 $T=\mathbb R^l$9, and there exists an $\|\cdot\|$0-continuous spectral random field with the same finite-dimensional distributions (Hashorva, 24 Jul 2025).

The tail random field is then

$\|\cdot\|$1

where $\|\cdot\|$2 is independent and Pareto-type. A fundamental identity is

$\|\cdot\|$3

This identity links shifted tail fields and local tail fields and is central in the theory of regularly varying stationary random fields (Hashorva, 24 Jul 2025).

The 2021 homogeneous-class framework formulates the same circle of ideas with a general $\|\cdot\|$4-normalization. In that setting the local field $\|\cdot\|$5 satisfies a generalized time-change formula, while $\|\cdot\|$6 is characterized by a scaling identity and the condition that $\|\cdot\|$7 is Pareto. The two formulations agree in the special case $\|\cdot\|$8, which recovers the classical spectral tail process normalization (Hashorva, 2021).

Object	Definition	Role
$\|\cdot\|$9	jointly measurable representor	generates the class $\mathbb R^d$0
$\mathbb R^d$1 or $\mathbb R^d$2	$\mathbb R^d$3 under the tilted law	local or spectral shape at the origin
$\mathbb R^d$4	$\mathbb R^d$5	tail random field
$\mathbb R^d$6	cluster random field	random-shift generator of the same class

5. Random-shift constructions and cluster random fields

The jointly measurable theory uses $\mathbb R^d$7 and a random shift $\mathbb R^d$8 with density $\mathbb R^d$9 to construct new elements of $\alpha$00. When $\alpha$01 almost surely, the paper defines a random-shift representor $\alpha$02 and proves that $\alpha$03; it also introduces a local-field version $\alpha$04 and a further variant $\alpha$05 based on local cluster structure when $\alpha$06 may vanish. These constructions show that the class is not only closed under deterministic shifts but also rich in explicit normalized random-shift representors (Hashorva, 24 Jul 2025).

The 2022 cluster-field framework makes this construction systematic. A random field $\alpha$07 is a cluster random field if

$\alpha$08

and

$\alpha$09

Given an independent $\alpha$10-valued random shift $\alpha$11 with positive density $\alpha$12, the random-shift transform is

$\alpha$13

The class generated by such transforms is shift-invariant; conversely, if $\alpha$14 is a cluster random field, the resulting class is shift-invariant and purely dissipative, and every element has finite positive total $\alpha$15-mass,

$\alpha$16

The class does not depend on the particular choice of the positive-density shift variable $\alpha$17 (Hashorva, 2022).

Cluster random fields unify the original representor $\alpha$18, the spectral tail field $\alpha$19, and the tail field $\alpha$20 within a single random-shift representation. The 2022 paper gives explicit constructions of $\alpha$21 from each of these objects, including anchored and shift-involution-based versions. It also proves that when $\alpha$22, the induced tail measure admits a cluster/random-shift form and the associated stationary max-stable field has a Rosiński / mixed moving maxima representation

$\alpha$23

Thus the finite total cluster mass condition is equivalent to purely dissipative behavior and to the existence of a random-shift representation (Hashorva, 2022).

6. Broader homogeneous-class theory and related models

The 2021 theory places shift-generated classes inside a more general family of $\alpha$24-homogeneous equivalent classes determined by a measurable map

$\alpha$25

For a random field $\alpha$26, the class $\alpha$27 consists of all representers with the same structural identities and the same tail measure

$\alpha$28

Equality of classes is therefore equality of tail measures, not merely similarity of path properties. When $\alpha$29 is an additive group, a class is shift-invariant if

$\alpha$30

and then random shifts preserve the class: $\alpha$31 for every independent $\alpha$32-valued random variable $\alpha$33 (Hashorva, 2021).

This general framework introduces universal maps $\alpha$34, including

$\alpha$35

and

$\alpha$36

which are used to study dissipativity, conservativity, maximal indices, and extremal asymptotics. For max-stable fields, when $\alpha$37 has nonnegative components, the class $\alpha$38 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$39-stable random fields (Hashorva, 2021).

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 $\alpha$40-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 (Hashorva, 24 Jul 2025).