## Abstract

We investigate the large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate (S_{n})_{n}_{∈N} =^{(} functions n^{−1}(^{∑}n k=1^{X}k,^{∑}for the sequence of two-dimensional random vectors n. Via the Contraction Principle, we provide k=1^{X2}k^{))}n∈N the explicit rate functions for the sample mean and the sample second moment. In the AR(1) case, we also give the explicit rate function for the sequence of two-dimensional (W_{n})_{n}_{⩾2} =^{(}n^{−1}(^{∑}n k=1^{X2}k^{,∑}n random vectors k=2^{X}kX_{k−1})^{)}, but we obtain an n⩾2 analytic rate function that gives different values for the upper and lower bounds, depending on the evaluated set and its intersection with the respective set of exposed points. A careful analysis of the properties of a certain family of Toeplitz matrices is necessary. The large deviations properties of three particular sequences of one-dimensional random variables will follow after we show how to apply a weaker version of the Contraction Principle for our setting, providing new proofs for two already known results on the explicit deviation function for the sample second moment and Yule-Walker estimators. We exhibit the properties of the large deviations of the first-order empirical autocovariance, its explicit deviation function and this is also a new result.

Original language | English (US) |
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Pages (from-to) | 177-212 |

Number of pages | 36 |

Journal | Probability, Uncertainty and Quantitative Risk |

Volume | 8 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

## Keywords

- Autoregressive processes
- Empirical autocovariance
- Large deviations
- Moving average processes
- Sample moments
- Toeplitz matrices
- Yule-Walker estimator

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty
- Applied Mathematics