Stable random variables: Convolution and reliability

Otiniano, Cira E. G.; Sousa, Thiago R.; Rathie, Pushpa N.

doi:10.1016/j.cam.2012.10.013

Cited by 10 publications

(3 citation statements)

References 5 publications

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“…Convolution integral (23) has been studied in a number of papers as fundamental solution of double-order space-fractional diffusion equation [9], as generalised Voigt function [40,56], or as a sum of two independent stable random variables [53,51].…”

Section: Power-law Tails Zero-size Jumps and Self-similaritymentioning

confidence: 99%

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Pagnini,

Vitali

2021

Preprint

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We study Markovian continuous-time random walk models for Lévy flights and we show an example in which the convergence to stable densities is not guaranteed when jumps follow a bi-modal power-law distribution that is equal to zero in zero. The significance of this result is two-fold: i) with regard to the probabilistic derivation of the fractional diffusion equation and also ii) with regard to the concept of site fidelity in the framework of Lévy-like motion for wild animals.

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Section: Power-law Tails Zero-size Jumps and Self-similaritymentioning

confidence: 99%

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Pagnini,

Vitali

2021

Preprint

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“…Such a convolution is sometimes called the Voigt profile, see related discussions in Ref. [55]. The scaling behavior of…”

Section: Convolution Of Lévy and Gaussian Distributionmentioning

confidence: 99%

Fractional advection-diffusion-asymmetry equation

Wang,

Barkai

2020

Preprint

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Fractional kinetic equations employ non-integer calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations we propose and investigate a fractional advection-diffusion equation describing the biased spreading. While usual transport is described by diffusion and drift, we find a third term describing symmetry breaking which is omnipresent for transport in disordered systems. Our work is based on continuous time random walks with a finite mean waiting time and a diverging variance, a case that on the one hand is very common and on the other was missing in the kaleidoscope literature of fractional equations. The fractional space derivative stems from long trapping times while previously it was interpreted as a consequence of spatial Lévy flights.

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“…Under these assumptions, all ρ i (x) belong to the same domain of attraction of a stable distribution function. In this case, the distribution of the sum j a ij x j can be obtained from the generalized central limit theorem [10], which leads to…”

Section: Theorymentioning

confidence: 99%

Power-law exponent of the Bouchaud-Mézard model on regular random networks

Ichinomiya

2013

Phys. Rev. E

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We study the Bouchaud-Mézard model on a regular random network. By assuming adiabaticity and independency, and utilizing the generalized central limit theorem and the Tauberian theorem, we derive an equation that determines the exponent of the probability distribution function of the wealth as x→∞. The analysis shows that the exponent can be smaller than 2, while a mean-field analysis always gives the exponent as being larger than 2. The results of our analysis are shown to be in good agreement with those of the numerical simulations.

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Stable random variables: Convolution and reliability

Cited by 10 publications

References 5 publications

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Should I stay or should I go? Zero-size jumps in random walks for Lévy flights

Fractional advection-diffusion-asymmetry equation

Power-law exponent of the Bouchaud-Mézard model on regular random networks

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