一类近不稳定过程的一致性与渐近正态性
Consistency and asymptotic normality in a class of nearly unstable processes
AI总结：
本文研究稳定但近不稳定自回归过程的推断问题，通过时变伴随矩阵建立框架，在实谱及复特征值假设下证明了经验协方差与OLS估计的一致性和渐近正态性，并通过模拟验证了渐近行为。
摘要：
本文研究一类稳定但近不稳定过程中的推断问题。考虑自回归过程，其中稳定与不稳定之间的桥梁由时变伴随矩阵 $A_{n}$ 表达，其谱半径 $ρ(A_{n}) < 1$ 且满足 $ρ(A_{n}) \rightarrow 1$。该框架通过聚焦单位圆的内部边界，特别适合理解单位根问题。在 $A$（即 $A_n$ 的极限）具有实谱的适当假设下，确立了经验协方差和 OLS 估计的一致性及渐近正态性，并推导了 $A$ 也包含复特征值的特定情形。渐近过程整合了一个单位根（位于 1 或 $-1$），甚至两个分别位于 1 和 $-1$ 的单位根。最后，一组模拟说明了 OLS 的渐近行为。结果主要通过 $L^2$ 计算和鞅三角阵列的极限理论证明。
英文摘要：
This paper deals with inference in a class of stable but nearly-unstable processes. Autoregressive processes are considered, in which the bridge between stability and instability is expressed by a time-varying companion matrix $A_{n}$ with spectral radius $ρ(A_{n}) < 1$ satisfying $ρ(A_{n}) \rightarrow 1$. This framework is particularly suitable to understand unit root issues by focusing on the inner boundary of the unit circle. Consistency is established for the empirical covariance and the OLS estimation together with asymptotic normality under appropriate hypotheses when $A$, the limit of $A_n$, has a real spectrum, and a particular case is deduced when $A$ also contains complex eigenvalues. The asymptotic process is integrated with either one unit root (located at 1 or $-1$), or even two unit roots located at 1 and $-1$. Finally, a set of simulations illustrate the asymptotic behavior of the OLS. The results are essentially proved by $L^2$ computations and the limit theory of triangular arrays of martingales.
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