Explicit bivariate rate functions for large deviations in AR(1) and MA(1) processes with Gaussian innovations

Maicon J. Karling, Artur O. Lopes, Sílvia R.C. Lopes*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 (Sn)n∈N =( functions n−1(n k=1Xk,for the sequence of two-dimensional random vectors n. Via the Contraction Principle, we provide k=1X2k))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 (Wn)n⩾2 =(n−1(n k=1X2k,∑n random vectors k=2XkXk−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 languageEnglish (US)
Pages (from-to)177-212
Number of pages36
JournalProbability, Uncertainty and Quantitative Risk
Issue number2
StatePublished - 2023


  • 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


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