{"publication":"Electronic Journal of Probability","title":"Persistence of autoregressive sequences with logarithmic tails","date_created":"2023-01-10T08:28:12Z","page":"1-43","type":"journal_article","language":[{"iso":"eng"}],"intvolume":"        27","_id":"35650","year":"2022","publication_status":"published","date_updated":"2023-01-10T08:29:02Z","status":"public","author":[{"last_name":"Denisov","full_name":"Denisov, Denis","first_name":"Denis"},{"first_name":"Günter","full_name":"Hinrichs, Günter","last_name":"Hinrichs"},{"first_name":"Martin","id":"48880","full_name":"Kolb, Martin","last_name":"Kolb"},{"last_name":"Wachtel","full_name":"Wachtel, Vitali","first_name":"Vitali"}],"doi":"https://doi.org/10.48550/arXiv.2203.14772","abstract":[{"text":"We consider autoregressive sequences Xn = aXn−1 + ξn and\r\nMn = max{aMn−1 , ξn} with a constant a ∈ (0, 1) and with positive, in-\r\ndependent and identically distributed innovations {ξk }. It is known that if\r\nP(ξ1 > x) ∼ d\r\nlog x with some d ∈ (0, − log a) then the chains {Xn} and {Mn}\r\nare null recurrent. We investigate the tail behaviour of recurrence times in this\r\ncase of logarithmically decaying tails. More precisely, we show that the tails\r\nof recurrence times are regularly varying of index −1 − d/ log a. We also prove\r\nlimit theorems for {Xn} and {Mn} conditioned to stay over a fixed level x0.\r\nFurthermore, we study tail asymptotics for recurrence times of {Xn} and {Mn}\r\nin the case when these chains are positive recurrent and the tail of log ξ1 is\r\nsubexponential.","lang":"eng"}],"volume":27,"publisher":"Institute of Mathematical Statistics","citation":{"bibtex":"@article{Denisov_Hinrichs_Kolb_Wachtel_2022, title={Persistence of autoregressive sequences with logarithmic tails}, volume={27}, DOI={<a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>}, journal={Electronic Journal of Probability}, publisher={Institute of Mathematical Statistics}, author={Denisov, Denis and Hinrichs, Günter and Kolb, Martin and Wachtel, Vitali}, year={2022}, pages={1–43} }","chicago":"Denisov, Denis, Günter Hinrichs, Martin Kolb, and Vitali Wachtel. “Persistence of Autoregressive Sequences with Logarithmic Tails.” <i>Electronic Journal of Probability</i> 27 (2022): 1–43. <a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>.","ieee":"D. Denisov, G. Hinrichs, M. Kolb, and V. Wachtel, “Persistence of autoregressive sequences with logarithmic tails,” <i>Electronic Journal of Probability</i>, vol. 27, pp. 1–43, 2022, doi: <a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>.","ama":"Denisov D, Hinrichs G, Kolb M, Wachtel V. Persistence of autoregressive sequences with logarithmic tails. <i>Electronic Journal of Probability</i>. 2022;27:1-43. doi:<a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>","apa":"Denisov, D., Hinrichs, G., Kolb, M., &#38; Wachtel, V. (2022). Persistence of autoregressive sequences with logarithmic tails. <i>Electronic Journal of Probability</i>, <i>27</i>, 1–43. <a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>","short":"D. Denisov, G. Hinrichs, M. Kolb, V. Wachtel, Electronic Journal of Probability 27 (2022) 1–43.","mla":"Denisov, Denis, et al. “Persistence of Autoregressive Sequences with Logarithmic Tails.” <i>Electronic Journal of Probability</i>, vol. 27, Institute of Mathematical Statistics, 2022, pp. 1–43, doi:<a href=\"https://doi.org/10.48550/arXiv.2203.14772\">https://doi.org/10.48550/arXiv.2203.14772</a>."},"user_id":"85821","department":[{"_id":"96"}]}