对称稳定自回归序列的近单位根持续性
Near-unit-root persistence of symmetric and asymmetric stable autoregressive sequences
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中文总结 AI 辅助
该研究探讨对称稳定AR(1)序列在自回归系数趋近1时的近单位根持续性，推导其指数持续性率的上下界，明确其渐近行为并简化为平稳稳定奥恩斯坦-乌伦贝克过程的密采样问题。
中文摘要 AI 辅助
当自回归系数趋近于1时，持续性会发生特征变化：对于每个固定的0<a<1，零以上的生存概率呈指数衰减，而在单位根对称随机游走的情况下，生存概率为n^{-1/2}量级。我们研究由对称α-稳定新息驱动的AR(1)序列的这种转变，并将其指数持续性率记为Λ(a,α)。整个序列可通过在几何扩展时间网格上观测的单个稳定莱维过程获得精确表示。与连续半直线生存概率的比较给出Λ(a,α) ≤ (α/2)log(1/a)。对于0<α<2，该界否定了Hinrichs、Kolb和Wachtel关于正则变化新息尾的猜想的稳定特例。为获得相同近单位量级的下界，我们结合稳定过程对子采样的封闭性与单调性耦合，证明当a↑1时Λ(a,α) ≍ log(1/a)，且比值Λ(a,α)/log(1/a)收敛于(0,α/2]中的一个极限，该极限等于其在0<a<1上的上确界。最后，Lamperti变换将该常数的识别简化为平稳稳定奥恩斯坦-乌伦贝克过程的密采样持续性问题，现有高斯理论确定了α=2时的精确值；对于0<α<2，识别该值需要控制路径在零以下穿过并在连续观测间返回零以上的情况。
英文摘要
For an AR($1$) sequence with coefficient $0 < a < 1$, the probability of staying above zero decays exponentially in the number of steps; at the unit root $a=1$ it decays polynomially, with order $n^{-1/2}$ for symmetric increments. We study this transition for AR($1$) sequences driven by symmetric and asymmetric $α$-stable innovations. We write $Λ(a,α)$ for the symmetric exponential persistence rate and $Λ(a,α,ρ)$ for the rate of a strictly stable law with positivity parameter $ρ$. The entire chain admits an exact representation through a single stable Lévy process observed on a geometrically expanding time grid. In the symmetric case, comparison with continuous half-line survival gives $Λ(a,α) \leq \fracα{2}\log{(1/a)}$. For $0 < α< 2$, this bound disproves the stable specialization of a conjecture of Hinrichs, Kolb and Wachtel concerning regularly varying innovation tails. Combining stable closure under subsampling with a monotonicity coupling yields a lower bound of the same near-unit order. This proves $Λ(a,α) \asymp \log{(1/a)}$ as $a \uparrow 1$ and shows that $Λ(a,α)/\log{(1/a)}$ converges to a limit in $(0,α/2]$ equal to the supremum of that ratio over $0 < a < 1$. A Lamperti transformation reduces identification of this constant to a dense-sampling problem for a stationary stable Ornstein-Uhlenbeck process. Existing Gaussian theory determines the sharp value at $α=2$; for $0 < α< 2$, rescued crossings between observations remain the obstacle. For asymmetric strictly stable innovations with $α\neq 1$ and admissible $ρ\in(0,1)$, the corresponding bounds are $Λ(a,α,ρ) \leq α(1-ρ)\log{(1/a)}$ and $Λ(a,α,ρ) \asymp \log{(1/a)}$. At $α=2$, only the symmetric case $ρ=1/2$ occurs.
发表机构
- Universidade de São Paulo（圣保罗大学）
- King Fahd University of Petroleum and Minerals（法赫德国王石油与矿产大学）
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