Integer-Valued Multifractal Processes

• Integer-valued multifractal processes are count-valued stochastic processes that preserve integrality through discrete scaling operations like thinning.
• They extend classical multifractal frameworks by using compound Poisson and Cox models to incorporate multifractal scaling into event count data.
• Factorial moments in these processes exhibit exact multifractal scaling, enabling precise inference and empirical analysis of scale-dependent heterogeneity.

Searching arXiv for the cited papers to ground the article in current records. Integer-valued multifractal processes are count-valued stochastic processes whose scale dependence is governed by a nonlinear scaling function, but whose scaling mechanism is formulated so that integrality is preserved. In the current literature, the topic has two closely related strands. One is a direct discrete framework in which thinning replaces scalar multiplication and integer-valued multifractal processes are constructed by time changing compound Poisson processes with nondecreasing multifractal clocks (Grahovac, 26 Sep 2025). The other models observed event counts as Poisson or Cox processes with multifractal latent intensities, so that the counts are integer-valued while their box-counting statistics inherit multifractal scaling from the underlying random measure (Baïle et al., 2020). Both strands sit within the broader theory of multifractal processes defined by random scaling factors, which generalizes self-similarity and links multifractal scaling to infinitely divisible laws and Lévy-type constructions (Grahovac, 2020).

1. Conceptual foundations

The general multifractal paradigm starts from the scaling of moments

$E|X(t)|^q = c(q)\, t^{\tau(q)},$

where nonlinearity of $\tau(q)$ in $q$ is the hallmark of multifractality (Grahovac, 2020). In the general definition of a multifractal process, one replaces deterministic self-similar rescaling by a random scaling family: $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$ with positive random variables $M(\lambda,t)$ that are independent of the process and identically distributed in $t$ for fixed $\lambda$ (Grahovac, 2020). This formulation includes self-similar processes as the special case where $M(\lambda,t)$ is deterministic.

For integer-valued processes, ordinary random multiplication does not preserve the state space $\mathbb{Z}_+$. The central problem is therefore not only to exhibit nonlinear moment scaling, but to do so under an algebra compatible with count data. The 2025 framework addresses this by introducing thinning as the discrete counterpart to scalar multiplication and by formulating multifractal scaling directly in terms of thinning-based random rescaling (Grahovac, 26 Sep 2025).

The general theory also imposes structural constraints on admissible scaling factors. The multiplicative structure

$$M(\lambda_1 \lambda_2, t) \overset{d}{=} M(\lambda_1,\lambda_2 t)\cdot M(\lambda_2,t)$$

implies infinite divisibility of $\tau(q)$0, and for suitable scale domains the marginals of $\tau(q)$1 are those of a Lévy process (Grahovac, 2020). This is important because many multifractal clocks and intensities used in count models are built from infinitely divisible cascades or related Lévy-driven constructions.

2. Discrete scaling through thinning

The discrete framework replaces scalar multiplication by thinning. For deterministic $\tau(q)$2, Steutel–van Harn binomial thinning is defined by

$\tau(q)$3

with probability generating function

$\tau(q)$4

This operation preserves integrality and gives a natural notion of random contraction for count variables (Grahovac, 26 Sep 2025).

The construction extends to more general thinning schemes through a semigroup of pgfs $\tau(q)$5. In that setting,

$\tau(q)$6

and for a random multiplier $\tau(q)$7 independent of $\tau(q)$8,

$\tau(q)$9

This generalization is the discrete analogue of multiplying a real-valued random variable by a random factor (Grahovac, 26 Sep 2025).

Within this framework, the multifractal property for a count process takes the form

$q$0

which is the direct integer-valued counterpart of the classical relation

$q$1

The significance of this step is conceptual as well as technical: it provides a scaling operation native to count processes rather than importing real-valued multiplication into a discrete setting where it is not closed (Grahovac, 26 Sep 2025).

3. Principal constructions

The main direct construction begins with a compound Poisson process

$q$2

where $q$3 is a standard Poisson process and the jump sizes $q$4 are iid with pgf $q$5. Its pgf is

$q$6

One then chooses a nondecreasing multifractal clock $q$7 satisfying the classical multifractal scaling law

$q$8

and defines

$q$9

The resulting process is integer-valued and satisfies

$$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$0

thereby realizing a genuine integer-valued multifractal process (Grahovac, 26 Sep 2025).

A second construction arises in spatial point process theory. In a Cox process, or doubly stochastic Poisson process, the count in a region $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$1 obeys

$$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$2

where the intensity measure $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$3 is random. When $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$4 is a multifractal random measure, the observed cell counts are integer-valued, but their statistics across scales inherit the multifractal properties of the latent intensity (Baïle et al., 2020). This gives a broad class of multifractal spatial point patterns.

The broader literature on multifractal products supplies further sources of latent multifractal fields. For a stationary positive process $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$5 with $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$6 and scaling parameter $$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$7, the finite product

$$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$8

and its cumulative version

$$\{ X(\lambda t) \}_{t \in \mathcal{S}} \overset{d}{=} \{ M(\lambda, t) X(t) \}_{t \in \mathcal{S}},$$9

converge under explicit $M(\lambda,t)$0 conditions and produce limiting processes with nonlinear moment and Rényi functions (Denisov et al., 2011). This suggests a natural source of multifractal clocks or intensities for count models, even though the products themselves are positive real-valued rather than integer-valued.

4. Scaling laws, moments, and spectra

For integer-valued multifractal processes of the form $M(\lambda,t)$1, the moments are not generally pure power laws. If $M(\lambda,t)$2 is multifractal with scaling function $M(\lambda,t)$3, then for integer $M(\lambda,t)$4 such that $M(\lambda,t)$5,

$M(\lambda,t)$6

for explicit constants $M(\lambda,t)$7 (Grahovac, 26 Sep 2025). In the special case where $M(\lambda,t)$8 is a standard Poisson process with unit jumps,

$M(\lambda,t)$9

with Stirling numbers of the second kind. This decomposition explains why raw moments of count processes may deviate from exact power-law scaling even when the clock is multifractal.

Factorial moments provide the exact discrete analogue of moment scaling. For the same construction,

$t$0

where

$t$1

The paper emphasizes that factorial moments do exhibit exact multifractal scaling, and that using factorial moments is important for instance when estimating the scaling function from count data (Grahovac, 26 Sep 2025).

In spatial multifractal point processes, the central objects are partition functions and empirical moments of the latent intensity: $t$2

$t$3

The singularity spectrum is defined by

$t$4

with Legendre transform

$t$5

Strict concavity of $t$6 indicates multifractality, whereas linearity would indicate simple monofractal behavior (Baïle et al., 2020).

Several canonical examples illustrate these formulas. For a log-normal multiplicative cascade used as clock,

$t$7

while in synthetic log-normal cascade tests of the spatial methodology the estimated spectra match the theoretical parabolic forms

$t$8

(Grahovac, 26 Sep 2025, Baïle et al., 2020).

5. Relation to general multifractal process theory

The discrete theory inherits much of its structure from the general theory of multifractal processes. A central result of the general framework is that the scaling family has multiplicative and infinitely divisible structure, and that on $t$9 the moment scaling can be written as

$\lambda$0

where $\lambda$1 is the Laplace exponent of the background Lévy process (Grahovac, 2020). This gives a broad template for multifractal laws beyond classical cascades.

One systematic construction is the Lamperti-inspired class

$\lambda$2

built from a Lévy process $\lambda$3 and a stationary process $\lambda$4. Its scaling factors are

$\lambda$5

This establishes a flexible route from Lévy and stationary processes to multifractal processes on $\lambda$6 (Grahovac, 2020).

For integer-valued models, however, the same paper notes a basic obstruction: exponentiating Lévy processes typically yields positive real-valued processes, not integer-valued ones. An additional step is needed, such as embedding the multifractality into an arrival intensity of an integer-valued process or reinterpreting multiplicative scaling through thinning (Grahovac, 2020). The later thinning-based framework can be read as a direct resolution of that obstruction (Grahovac, 26 Sep 2025).

The literature on multifractal products broadens the catalog of possible multifractal drivers. Under conditions such as $\lambda$7, suitable monotonicity of the joint moment functions, and summability conditions on covariance decay, cumulative products converge in $\lambda$8 to non-degenerate multifractal limit processes (Denisov et al., 2011). The corresponding scaling exponents satisfy

$\lambda$9

and the Rényi function obeys

$M(\lambda,t)$0

for processes with exponentially decaying correlations (Denisov et al., 2011). Log-normal, log-gamma, log-tempered stable, log-normal tempered stable, and log-variance gamma scenarios are all treated explicitly.

6. Inference and empirical manifestations

Inference is substantially more difficult than construction because the latent multifractal driver is usually unobserved. In the spatial Cox framework, only the point pattern is observed, not the underlying intensity $M(\lambda,t)$1. The proposed solution is to approximate the distribution of $M(\lambda,t)$2 at each scale $M(\lambda,t)$3 by a finite mixture

$M(\lambda,t)$4

which induces an observed count distribution

$M(\lambda,t)$5

The parameters are then estimated by maximum likelihood through a standard Expectation-Maximization procedure, and the moments are reconstructed from the fitted mixture (Baïle et al., 2020).

Applied to annual wildfire ignition data in the three French Mediterranean regions Corsica, PACA, and LR, this methodology yields a well defined scaling behavior for the moments at each order $M(\lambda,t)$6, with a non-linear spectrum of scaling exponents $M(\lambda,t)$7 (Baïle et al., 2020). The extracted spectra are strictly concave, the singularity spectrum is non-trivial, and the analysis is confirmed by a direct spatial correlation estimation of the intensity logarithms whose slowly decreasing shape corresponds to the hallmark of multifractal cascades. The multifractal features appear to be constant over time and similar over the three regions.

The reported parameter values for the wildfire application are also informative: the fractal dimension is $M(\lambda,t)$8, the intermittency coefficient is $M(\lambda,t)$9, and the support of $\mathbb{Z}_+$0 ranges from about $\mathbb{Z}_+$1 to $\mathbb{Z}_+$2 (Baïle et al., 2020). The paper further states that the observed clustering of wildfire occurrences is explained almost entirely by spatial variations in the intensity field, not by direct correlations between event locations, as confirmed by inhomogeneous Ripley $\mathbb{Z}_+$3 analysis. A plausible implication is that, in some spatial count systems, much of the apparent interaction structure can be reinterpreted as heterogeneity in a multifractal risk surface rather than direct dependence among events.

7. Interpretation, misconceptions, and scope

A common misconception is that integer-valued multifractality is obtained simply by taking a classical multifractal process and forcing its values to be integers. The available theory points in a different direction. Because the scaling mechanism of classical multifractals is multiplicative on the positive reals, genuine integer-valued models require either a discrete scaling operation such as thinning or an indirect construction in which the multifractal object is a latent clock or intensity and the observed process is Poisson or Cox (Grahovac, 26 Sep 2025, Grahovac, 2020).

A second misconception is that ordinary moments alone always reveal the multifractal law of count processes. In the thinning-based framework, raw moments of $\mathbb{Z}_+$4 are sums of powers $\mathbb{Z}_+$5 and therefore need not obey exact power-law scaling; by contrast, factorial moments inherit the clock’s scaling exactly (Grahovac, 26 Sep 2025). For empirical work on counts, this distinction is methodological rather than merely notational.

A third misconception is that multifractal count data can be summarized adequately by counts per unit area or time. The wildfire study explicitly concludes that the annual ignition risk cannot be reduced to providing a number of events per $\mathbb{Z}_+$6, because the relevant structure lies in the non-trivial singularity spectrum and in the scale dependence of higher-order moments (Baïle et al., 2020). More generally, the topic concerns scale-dependent heterogeneity, intermittency, and clustering strength, not only mean intensity.

Taken together, the literature defines integer-valued multifractal processes as a technically coherent extension of multifractal theory to count data. The direct thinning-based approach supplies the discrete scaling principle; time-changed compound Poisson models provide an explicit class of examples; Cox and Poisson models with multifractal intensities show how the same ideas manifest in spatial event data; and general multifractal process theory supplies the Lévy, cascade, and product structures from which clocks and intensities can be built (Grahovac, 26 Sep 2025, Baïle et al., 2020, Grahovac, 2020, Denisov et al., 2011).