Abstract
In this article we study a small random perturbation of a linear recurrence equation. If all the roots of its corresponding characteristic equation have modulus strictly less than one, the random linear recurrence goes exponentially fast to its limiting distribution in the total variation distance as time increases. By assuming that all the roots of its corresponding characteristic equation have modulus strictly less than one and rather mild conditions, we prove that this convergence happens as a switch-type, i.e., there is a sharp transition in the convergence to its limiting distribution. This fact is known as a cut-off phenomenon in the context of stochastic processes.
Acknowledgments
G. Barrera gratefully acknowledges support from a post-doctorate Pacific Institute for the Mathematical Sciences (PIMS, 2017–2019) grant held at the Department of Mathematical and Statistical Sciences at University of Alberta. S. Liu is sincerely grateful to his advisor, Yingfei Yi, for the supporting of his graduate research. Both authors would like to express their gratitude to University of Alberta for all the facilities used along the realization of this work. The authors would like to thank the constructive and useful suggestions provided by the referees.
Citation
Gerardo Barrera. Shuo Liu. "A switch convergence for a small perturbation of a linear recurrence equation." Braz. J. Probab. Stat. 35 (2) 224 - 241, May 2021. https://doi.org/10.1214/20-BJPS474
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