We consider a stable but nearly unstable autoregressive process of any order. The bridge between stability and instability is expressed by a time-varying companion matrix An with spectral radius ρ(An) < 1 satisfying ρ(An)→ 1. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of An through 1−ρ(An) and, as a by-product, we establish a moderate deviation principle for the OLS estimator when Γ, the renormalized asymptotic variance of the process, is invertible. Finally, when Γ is singular, we also provide a compromise in the form of a moderate deviation principle for a penalized version of the estimator. Our proofs essentially rely on troncations and m–dependent sequences with unbounded m.

中文翻译：

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一类稳定但几乎不稳定的过程中的中等偏差

我们考虑任何阶的稳定但几乎不稳定的自回归过程。稳定与不稳定之间的桥梁由时变伴随矩阵 An 表示，其谱半径 ρ(An) < 1 满足 ρ(An)→ 1。在该框架中，我们建立了经验协方差的适度偏差原则，仅依赖于An 的元素到 1−ρ(An) 并且作为副产品，当 Γ（过程的重归一化渐近方差）可逆时，我们为 OLS 估计量建立了适度偏差原则。最后，当 Γ 是奇异的时，我们还以适度偏差原则的形式为估计量的惩罚版本提供折衷方案。我们的证明本质上依赖于截断和具有无限 m 的 m 相关序列。