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Linear combinations of i.i.d. strictly stable variables with random coefficients and their application to anomalous diffusion processes

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  • Hottovy, Scott
  • Pagnini, Gianni

Abstract

We show that linear combinations of independent and identically distributed strictly stable variables with positive random coefficients is equal in distribution to a function of these random coefficients times a random variable from the same stable distribution. Furthermore, this result is used to show that a random linear combination of independent standard Wiener processes has the distribution of a function of these random coefficients times one standard Wiener process. In opposition to the central limit theorem, this result does not require a large number of terms but it holds with two or more terms. This has implications to simplify stochastic differential equations with a finite number of noises with random coefficients that can be used in modeling anomalous diffusion.

Suggested Citation

  • Hottovy, Scott & Pagnini, Gianni, 2024. "Linear combinations of i.i.d. strictly stable variables with random coefficients and their application to anomalous diffusion processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 647(C).
    • Handle: RePEc:eee:phsmap:v:647:y:2024:i:c:s0378437124004217
    DOI: 10.1016/j.physa.2024.129912
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    References listed on IDEAS

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    1. Janczura, Joanna & Burnecki, Krzysztof & Muszkieta, Monika & Stanislavsky, Aleksander & Weron, Aleksander, 2022. "Classification of random trajectories based on the fractional Lévy stable motion," Chaos, Solitons & Fractals, Elsevier, vol. 154(C).
    2. Giona, Massimiliano & Pezzotti, Chiara & Procopio, Giuseppe, 2024. "Another normality is possible. Distributive transformations and emergent Gaussianity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 634(C).
    3. Silvia Vitali & Iva Budimir & Claudio Runfola & Gastone Castellani, 2019. "The Role of the Central Limit Theorem in the Heterogeneous Ensemble of Brownian Particles Approach," Mathematics, MDPI, vol. 7(12), pages 1-9, November.
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