Report
Persistence of autoregressive sequences with logarithmic tails
| العنوان: | Persistence of autoregressive sequences with logarithmic tails |
|---|---|
| المؤلفون: | Denisov, Denis, Hinrich, Gunter, Kolb, Martin, Wachtel, Vitali |
| سنة النشر: | 2022 |
| المجموعة: | Mathematics |
| مصطلحات موضوعية: | Mathematics - Probability, Primary 60G50, Secondary 60G40, 60F17 |
| الوصف: | We consider autoregressive sequences $X_n=aX_{n-1}+\xi_n$ and $M_n=\max\{aM_{n-1},\xi_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{\xi_k\}$. It is known that if $\mathbf P(\xi_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\{M_n\}$ conditioned to stay over a fixed level $x_0$. Furthermore, we study tail asymptotics for recurrence times of $\{X_n\}$ and $\{M_n\}$ in the case when these chains are positive recurrent and the tail of $\log\xi_1$ is subexponential. Comment: 42 pages |
| نوع الوثيقة: | Working Paper |
| URL الوصول: | http://arxiv.org/abs/2203.14772 |
| رقم الانضمام: | edsarx.2203.14772 |
| قاعدة البيانات: | arXiv |
| الوصف غير متاح. |