Linear Fractional Stable Motion

• Linear Fractional Stable Motion (LFSM) is a self-similar, heavy-tailed stochastic process with stationary increments and long-range dependence.
• LFSM generalizes fractional Brownian motion by using an integral representation with an α-stable random measure to capture non-Gaussian behavior.
• Advanced estimation methods such as wavelet-based inference and power variations enable consistent parameter estimation despite infinite variance challenges.

Linear Fractional Stable Motion (LFSM) is a canonical class of self-similar, heavy-tailed stochastic processes that generalizes fractional Brownian motion (fBm) by allowing its increments to follow stable non-Gaussian laws. LFSM is characterized by stationary increments, self-similarity, and the presence of long-range dependence and heavy tails governed by its parameters. Its mathematical tractability, scaling properties, and capacity to interpolate between classical Gaussian processes and stable Lévy processes make LFSM fundamental to the theory and applications of stable stochastic modeling.

1. Mathematical Definition and Construction

LFSM, denoted $\{X_{H,\alpha}(t)\}_{t\in\mathbb{R}}$, is constructed as an integral with respect to an $\alpha$-stable random measure $M_\alpha$, using a linear fractional kernel. The canonical representation is: $$X_t = \int_{-\infty}^\infty \left\{ (t-u)_+^{H-1/\alpha} - (-u)_+^{H-1/\alpha} \right\} M_\alpha(du),$$ where $0 < H < 1$ is the Hurst parameter and $0 < \alpha < 2$ is the stability index. Here, $M_\alpha$ is a symmetric $\alpha$-stable (S$\alpha$S) random measure with Lebesgue control.

Key properties include:

• Self-similarity: $X_{at} \overset{fdd}{=} a^H X_t$ for all $a>0$.
• Stationary increments: The distribution of $X_{t+s} - X_s$ does not depend on $s$.
• Heavy tails: For $\alpha<2$, increments have power-law tails (no second moment).
• Fractional Brownian motion as special case: When $\alpha=2$, $X_t$ is (up to scaling) fBm.

LFSM is equivalently represented as a fractional integral of a stable Lévy process: $X_t = I^{H-1/\alpha} L_\alpha(t),$ where $I^{\eta}$ is the Riemann–Liouville fractional integral and $L_\alpha$ is symmetric $\alpha$-stable Lévy motion (Feltes et al., 2020).

2. Regularity and Multifractal Structure

The path regularity of LFSM is governed jointly by $H$ and $\alpha$. For $H>1/\alpha$, sample paths are (uniformly) Hölder continuous of any order strictly less than $H-1/\alpha$. Notably, for each compact interval $I$,

$$|X_t - X_s| = O(|t-s|^{H-1/\alpha}), \quad t,s \in I, \quad a.s.$$

This critical exponent $H-1/\alpha$ is optimal; LFSM sample paths almost surely belong to the corresponding critical Hölder space $\mathcal{C}^{H-1/\alpha}(I)$ (Ayache et al., 2016). In the multifractional extension, where $H(t)$ varies, the regularity becomes non-homogeneous and is almost surely given by $\min_{x\in I} H(x) - 1/\alpha$ over $I$.

Multifractal analysis uncovers deeper structure:

• Using the 2-microlocal formalism, the multifractal spectrum of LFSM is a translation of that of the underlying driving stable Lévy process. Specifically,

$$\sigma_{X, t}(s') = \sigma_{L, t}(s') + H - 1/\alpha,$$

leading to a multifractal spectrum

$$d_X(h, V) = \begin{cases} \alpha (h - H) + 1, & h \in [H - 1/\alpha, H],\ -\infty, & \text{otherwise}, \end{cases}$$

for open $V$ (Balança, 2013).

3. Statistical Inference and Estimation

Parameter estimation for LFSM, specifically the triplet $(\sigma, \alpha, H)$, has driven significant methodological development owing to its non-Gaussianity and infinite variance properties. Recent works establish consistent estimators and functional limit theorems for the parameters using both wavelet and increment-based statistics:

• Wavelet-based inference: For known $H$, the decay rate of maximal absolute wavelet coefficients $D_j$ at scale $j$ obeys

$D_j \asymp 2^{-j(H-1/\alpha)},$

leading to the consistent estimator

$$\widehat{1/\alpha}_j = H + \frac{\log D_j}{j\log 2},$$

which satisfies $\widehat{\alpha}_j \to \alpha$ almost surely (Ayache et al., 2013).

• Power variation and characteristic function methods: For general $(\sigma, \alpha, H)$, estimation leverages power variations, negative power variations (to handle infinite moments), and empirical characteristic functions:

$$\widehat{H}_n = \frac{1}{p} \log_2 \left[\frac{\sum |\Delta^2 X|^p}{\sum |\Delta^1 X|^p}\right],\quad \phi_n(t) = \frac{1}{n} \sum \cos(t n^H \Delta^k X),$$

with analytic expressions linking $\sigma$ and $\alpha$ to $\phi_n$ via inversion formulas. Joint estimators achieve asymptotic normality (or non-central stable limits, depending on increment order) (1802.06373, Ljungdahl et al., 2019).

• Mixed processes and high-frequency data: In composite models (e.g., superpositions of LFSMs, mixed-fractional regimes), estimating equations involving moment functionals and empirical characteristic functions retain consistency and asymptotic normality under sharp conditions. Smooth threshold techniques can isolate contributions of distinct process components (Mies et al., 2022).
• Deep learning approaches: LSTM networks, trained on fractional Brownian/Ornstein-Uhlenbeck data, can estimate the Hurst parameter for Gaussian-like processes, but underperform for LFSM due to differences in process law and the absence of efficient LFSM simulators. For LFSM, traditional inference currently remains superior (Boros et al., 3 Jan 2024).

4. Simulation and Approximation Schemes

The nontrivial kernel structure and infinite variance present challenges in simulating and approximating LFSM paths:

• Lattice/Riemann-sum methods: Approximate the integral definition by discretizing both the stable measure and the kernel, yielding convergence in finite-dimensional distributions even for integrands in $L_\alpha$ (Dombry et al., 2013).
• Random wavelet series: For both LFSM and multifractional analogues, expansions in biorthogonal wavelets facilitate strong convergence in function spaces and explicit control of path regularity (Ayache et al., 2013).
• Fast fBm generators: Deep learning applications rely on high-throughput fBm simulation, but no comparably efficient LFSM generator is presently available (Boros et al., 3 Jan 2024).

5. Ergodic, Dependence, and Scaling Properties

LFSM exhibits rich dependence structures:

• Long-range dependence: Captured via power-law decay of codifference or so-called generalized measures, as covariance does not exist for $\alpha<2$ (Feltes et al., 2020).
• Codifference: The appropriate measure of dependence, given for S$\alpha$S variables by

$$\text{CD}(X, Y) = \|X\|_\alpha^\alpha + \|Y\|_\alpha^\alpha - \|X - Y\|_\alpha^\alpha,$$

and reduces to covariance for $\alpha=2$. Codifference is essential for non-Gaussian prediction and for understanding ergodic properties (Garcin et al., 21 Jul 2025).

• Ergodic-class regimes: LFSM increments may be dissipative, null-conservative, or positive-conservative, depending on the nature of the kernel and subordination or time-randomization applied (Jung, 2011).

6. Prediction and Forecasting

Forecasting LFSM extends fBm methods to non-Gaussian settings:

• Covariance-based projection is not defined for LFSM due to infinite variance.
• Codifference-based forecasting: Uses the codifference matrix to formulate unique discrete-time decompositions; conditional expectation (for $\alpha>1$) or metric projection (otherwise) yields the forecast. For real data (e.g., high-frequency FX rates), this approach outperforms fBm-based methods in directional prediction accuracy (Garcin et al., 21 Jul 2025).

Notably, the codifference framework distinguishes between persistence, antipersistence, independence ($H\gtrless1/\alpha$), and reveals a fourth "selective memory" regime for low $\alpha$, characterized by the lasting influence of extreme jumps.

7. Applications and Extensions

LFSM is a foundational model in fields requiring both scale invariance and heavy tails:

• Finance: Explains empirical features of financial time series (e.g., power-law order flow, antipersistence in returns, volatility modeling with separate Joseph and Noah effects) (Gontis, 2021, Garcin et al., 21 Jul 2025).
• Nonparametric regression with nonstationary, heavy-tailed covariates: LFSM limits support uniform convergence in additive functionals, crucial for the theory of kernel regression with nonstationary data (Duffy, 2015).
• Network traffic, image analysis, geophysics: The process’ roughness, self-similarity, and stability of increments under aggregation are well-matched to empirical traffic and imaging data.

Extensions include:

• Random-time and multifractional variants: Allow time-dependent Hurst exponents, yielding non-homogeneous/locally random multifractal spectra and path regularity (Ayache et al., 2013, Ayache et al., 2013, Balança, 2013).
• Recursive constructions, subordination, and time-change: From random walks at random times to hierarchical models, LFSM is a natural limit object or building block for processes arising from discrete systems or subordinated random times (Jung et al., 2011, Jung, 2011).

Summary Table: Key Features and Results

Aspect	LFSM ($\alpha<2$)	fBm ($\alpha=2$)
Increment law	Symmetric $\alpha$-stable	Gaussian
Covariance	Not defined	Explicit, used for all inference
Codifference	Finite, fundamental	Reduces to covariance
Regularity	$H-1/\alpha$ Hölder	$H$ Hölder
Multifractal spectrum	Shift of Lévy spectrum	Deterministic, pointwise exponent
Parameter estimation	Power/wavelet variation, characteristic function, minimal contrast, deep learning (limited)	Input-output moments, ML, OLS
Simulation	Lattice/wavelet/FFT	Davies–Harte, Wood–Chan

LFSM, thus, offers a rigorous framework for the analysis, simulation, and application of stochastic phenomena exhibiting both long memory and heavy tails. Its mathematical structure enables extension and adaptation in a broad range of modern probabilistic modeling contexts.