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Universal Poisson-process limits for general random walks

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  • Eliazar, Iddo

Abstract

This paper considers ensembles of general, independent and identically distributed, random walks. Taking the ensemble-size to grow infinitely large, and also taking the running-time of the random walks to grow infinitely large, universal Poisson-process limits are obtained. Specifically, it is established that the positions of general linear random walks converge universally to Poisson processes, over the real line, with uniform and exponential intensities. And, it is established that the positions of general geometric random walks converge universally to Poisson processes, over the positive half-line, with harmonic and power intensities. Corollaries to these universal convergence results yield the extreme-value statistics of Gumbel, Weibull, and Frechet.

Suggested Citation

  • Eliazar, Iddo, 2018. "Universal Poisson-process limits for general random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 1160-1174.
    • Handle: RePEc:eee:phsmap:v:512:y:2018:i:c:p:1160-1174
    DOI: 10.1016/j.physa.2018.08.038
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    1. Lux, Thomas & Alfarano, Simone, 2016. "Financial power laws: Empirical evidence, models, and mechanisms," Chaos, Solitons & Fractals, Elsevier, vol. 88(C), pages 3-18.
    2. Aleksander Janicki & Aleksander Weron, 1994. "Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes," HSC Books, Hugo Steinhaus Center, Wroclaw University of Science and Technology, number hsbook9401, December.
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    Cited by:

    1. Min, Seungsik & Shin, Ki-Hong & Baek, Woonhak & Kim, Kyungsik & You, Cheol-Hwan & Lee, Dong-In & Yum, Seong Soo & Kim, Wonheung & Chang, Ki-Ho, 2020. "Dynamical behavior of combined detrended cross-correlation analysis methods in random walks and Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    2. Sánchez-Sánchez, Sergio & Cortés-Pérez, Ernesto & Moreno-Oliva, Víctor I., 2024. "Binomial vs. Poisson statistics: From a toy model to a stochastic model for radioactive decay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 643(C).

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