Mixed Fractional Information (MFI)

• Mixed Fractional Information (MFI) is a finite, computable measure that serves as an alternative to classical Fisher information for heavy-tailed symmetric α-stable distributions.
• It operationalizes the rate of relative entropy dissipation via a canonical stable interpolation, linking score functions, entropy dynamics, and MMSE analogues.
• MFI extends to mixed fractional Brownian motion models, offering practical insights into information behavior in long-memory and noisy environments.

Mixed Fractional Information (MFI) provides a rigorous, computable, and finite analogue of Fisher information for symmetric α-stable (SαS) distributions, where classical Fisher information diverges for tail parameters α < 2. Defined as the rate of relative entropy dissipation under a canonical stable interpolation and capturing differences between SαS scales, MFI establishes a new information-theoretic metric for heavy-tailed families, with links to entropy dynamics, score functions, and estimation theory. The concept further interacts with estimation in mixed fractional Brownian motion (mFBM) models, where the information structure exhibits distinct and sometimes degenerate behaviors dependent on the noise composition and self-similarity index.

1. MFI for Symmetric α-Stable Laws: Motivation and Definition

Classical Fisher information, which quantifies the local curvature of relative entropy and underpins Cramér–Rao bounds, is infinite for symmetric α-stable ($\alpha<2$) distributions due to heavy tails. To address this, MFI is constructed to measure the "rate of entropy dissipation" between two SαS distributions $g_v^{(\alpha)}$ and $g_s^{(\alpha)}$ (scale parameters $v, s > 0$). The Kullback–Leibler divergence between these densities is

$$D(v)\;=\;\int_{-\infty}^\infty g_v^{(\alpha)}(x)\,\ln\frac{g_v^{(\alpha)}(x)}{g_s^{(\alpha)}(x)}\,dx,$$

and the Mixed Fractional Information is defined as its derivative with respect to $v$: $\mathrm{MFI}\;=\;D'(v).$ This operationalizes MFI as the initial rate of relative entropy dissipation along the canonical interpolation $X_t=(1-t)^{1/\alpha}X_0+t^{1/\alpha}Z_s^{(\alpha)}$, with $X_0\sim g_v^{(\alpha)}$, $Z_s^{(\alpha)}\sim g_s^{(\alpha)}$ (Cook, 18 Apr 2025).

2. Equivalent Representations and Consistency Identity

An alternative, but equivalent, representation of MFI arises from a fractional analogue of the de Bruijn identity, evaluated at $g_v^{(\alpha)}$0. Writing $g_v^{(\alpha)}$1 for the Fisher score and $g_v^{(\alpha)}$2 for the MMSE-related interpolation score, one obtains

$g_v^{(\alpha)}$3

The central result demonstrates the equivalence of these representations: $g_v^{(\alpha)}$4 This "consistency identity" connects the entropy dissipation rate to score function differences and validates MFI’s coherence. The proof employs the SαS scaling identity, differentiation under the integral, and integration by parts (Cook, 18 Apr 2025).

3. Properties: Non-Negativity and Explicit Cases

That $g_v^{(\alpha)}$5 with equality only when $g_v^{(\alpha)}$6 implies $g_v^{(\alpha)}$7 is sign-consistent: $g_v^{(\alpha)}$8 for $g_v^{(\alpha)}$9 and $g_s^{(\alpha)}$0 for $g_s^{(\alpha)}$1, so MFI vanishes only when the scales agree. For the Cauchy case ($g_s^{(\alpha)}$2), explicit computation yields

$g_s^{(\alpha)}$3

A manifestly non-negative symmetric form is

$g_s^{(\alpha)}$4

These explicit forms anchor MFI as a computable, finite index for information carried between stable laws (Cook, 18 Apr 2025).

4. Computational Validation and Practical Considerations

Numerical simulations using adaptive quadrature, high-order finite differences, and log-derivative estimation confirm the consistency identity for general $g_s^{(\alpha)}$5, with $g_s^{(\alpha)}$6 relative accuracy for $g_s^{(\alpha)}$7 and machine precision in the Cauchy and Gaussian limits. These computations show the stability of MFI under mesh refinement, making it practical for applications where reliable entropy dissipation rates are required between stable distributions (Cook, 18 Apr 2025).

5. Theoretical Connections: Score, MMSE, and I-MMSE Analogues

MFI’s integral formulation involves the difference in Fisher scores and an MMSE-related interpolation score, linking the entropy dissipation rate directly to score and regression quantities central in estimation theory. The core identity

$g_s^{(\alpha)}$8

parallels the Gaussian I–MMSE relationship, suggesting that MFI serves as a fractional substitute for Fisher information in heavy-tailed settings, potentially underpinning new versions of functional inequalities (e.g., weighted log-Sobolev or Poincaré-type) and motivating further study of fractional I-MMSE relations (Cook, 18 Apr 2025).

6. Extensions: Mixed Fractional Information in mFBM Models

In mixed fractional Brownian motion (mFBM) models, the observed process is $g_s^{(\alpha)}$9, combining fractional Brownian motion ($v, s > 0$0) of Hurst parameter $v, s > 0$1 and independent Brownian noise ($v, s > 0$2). Under high-frequency observation, the information structure—sometimes referred to as "mixed-fractional information"—has distinct behaviors:

• For $v, s > 0$3, only a lower-triangular renormalization yields non-singular asymptotic Fisher information, due to collinearity in the likelihood scores for $v, s > 0$4. The Fisher information matrix degenerates as $v, s > 0$5 (Cai, 30 Dec 2025).
• For $v, s > 0$6, after orthogonalization and explicit projection, the mixed-fractional Fisher information matrix becomes diagonal with closed-form entries dependent on $v, s > 0$7 and $v, s > 0$8, extracted via spectral analysis of fGn Toeplitz matrices (Cai, 6 Jan 2026).

The interplay between these MFI forms and the classical Fisher information further highlights the necessity of fractional or mixed formulations in heavy-tailed or long-memory regimes.

7. Outlook and Implications

MFI provides a robust, non-divergent measure of information for SαS distributions, with a rigorous foundation in entropy dissipation under the α-stable interpolation and explicit links to estimation theory and information geometry. Future research directions include extension to non-symmetric stable laws, characterization of the regularity class $v, s > 0$9, and the development of functional inequalities tailored to heavy-tailed phenomena. MFI’s foundational status is further underscored by its ability to mediate between score-centric and entropy-centric viewpoints, thereby facilitating statistical inference and functional analytical studies in stable and mixed-fractional environments (Cook, 18 Apr 2025, Cai, 30 Dec 2025, Cai, 6 Jan 2026).