What question did this study set out to answer?

The aim is to establish a framework for understanding how probability distributions function as fixed points of a specific nonlinear operator.

May 8, 2026Open Access

Operator Theory of Distribution Dynamics (OTDD v2.0)

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Mathematical framework explores fixed points of nonlinear operators in probability distributions, indicating foundational dynamics.

The aim is to establish a framework for understanding how probability distributions function as fixed points of a specific nonlinear operator.
Proved existence of fixed points using Schauder's fixed-point theorem.
Derived a closed-form dominant eigenvalue and spectral gap ratio for a scale-invariant Gamma kernel.
Classified distribution tails based on growth order, covering various types from power-law to stretched-exponential.
The dominant eigenvalue λ₁(a; κ, β) = β⁻ᵃ · Γ(κ+a) / Γ(κ) was found, indicating essential spectral characteristics.
The tail classification highlights that multiple distribution types originate from a single operator family.
A critical boundary q = 3/2 correlates with the GPM–GARCH CLT breakdown condition.

László Tatai (2026) studied this question.

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May 8, 2026Open Access

View Full Paper

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Authors

Mathematical framework explores fixed points of nonlinear operators in probability distributions, indicating foundational dynamics.

The aim is to establish a framework for understanding how probability distributions function as fixed points of a specific nonlinear operator.
Proved existence of fixed points using Schauder's fixed-point theorem.
Derived a closed-form dominant eigenvalue and spectral gap ratio for a scale-invariant Gamma kernel.
Classified distribution tails based on growth order, covering various types from power-law to stretched-exponential.
The dominant eigenvalue λ₁(a; κ, β) = β⁻ᵃ · Γ(κ+a) / Γ(κ) was found, indicating essential spectral characteristics.
The tail classification highlights that multiple distribution types originate from a single operator family.
A critical boundary q = 3/2 correlates with the GPM–GARCH CLT breakdown condition.

László Tatai (2026) studied this question.

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