论文标题
与对数尾巴的自回旋序列的持久性
Persistence of autoregressive sequences with logarithmic tails
论文作者
论文摘要
我们考虑自动回归序列$ x_n = ax_ {n-1}+ξ_n$和$ m_n = \ max \ {am_ {n-1}，ξ_n\} $，带有常数$ a \ in（0,1）$，并带有积极的，积极的，积极的，独立的，独立的，独立的innovations $ \ \ \ \ {ξ_} $。众所周知，如果$ \ mathbf p（ξ_1> x）\ sim \ frac {d} {\ log x} $带有一些$ d \ in（0， - \ log a）$，则链条$ \ {x_n \} $ and $ \ \ \ \ \ {m_n \} $是null recurrent的。在这种情况下，我们研究了复发时间的尾巴行为，在对数衰减的尾巴上。更确切地说，我们表明，复发时间的尾巴定期改变索引$ -1-d/\ log a $。我们还证明了$ \ {x_n \} $和$ \ {m_n \} $的限制定理，以保持固定级别$ x_0 $。此外，我们研究了$ \ {x_n \} $的复发时间和$ \ {m_n \} $的尾部渐近学。
We consider autoregressive sequences $X_n=aX_{n-1}+ξ_n$ and $M_n=\max\{aM_{n-1},ξ_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{ξ_k\}$. It is known that if $\mathbf P(ξ_1>x)\sim\frac{d}{\log x}$ 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 $\logξ_1$ is subexponential.
