In this talk, we consider asymptotic properties for the quadratic functionals and associated ordinary least squares (OLS) estimator in the autoregressive process of order 1 with Gaussian noise. Deviation inequalities and Cramer-type moderate deviations are achieved in the explosive and mildly explosive frameworks. The main methods used in this paper consist of the deviation inequalities for multiple Wiener-Ito integrals, as well as the asymptotic analysis techniques. This is a joint work with Yilong WAN.

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