# Moderate deviations in a class of stable but nearly unstable processes

Abstract : 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 $A_{n}$ with spectral radius $\rho(A_{n}) < 1$ satisfying $\rho(A_{n}) \rightarrow 1$. In that framework, we establish a moderate deviation principle for the empirical covariance only relying on the elements of $A_{n}$ through $1-\rho(A_{n})$ and, as a by-product, we establish a moderate deviation principle for the OLS estimator when $\Gamma$, the renormalized asymptotic variance of the process, is invertible. Finally, when $\Gamma$ 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 $m_{n}$--dependent sequences with an unbounded rate $(m_{n})$.
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Journal articles
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https://hal.archives-ouvertes.fr/hal-02123384
Contributor : Frédéric Proïa <>
Submitted on : Wednesday, May 8, 2019 - 8:32:46 AM
Last modification on : Friday, March 13, 2020 - 10:00:13 AM

### Identifiers

• HAL Id : hal-02123384, version 1
• ARXIV : 1905.02618

### Citation

Frédéric Proïa. Moderate deviations in a class of stable but nearly unstable processes. Journal of Statistical Planning and Inference, Elsevier, 2020, 208, pp.66-81. ⟨hal-02123384⟩

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