Moderate deviations principle for empirical covariance from a unit root

In the present paper, we consider the linear autoregressive model in $\text{rr}$, $$X_{k,n}={\theta }_n{X}_{k,n-1}+{\xi }_k,k=0,1,\mathrm{.}\mathrm{.}\mathrm{.},n,n\ge 1$$ where $\theta_n\in [0,1)$isunknown,$(\xi_k)_{k\in\zz}$ is a sequence of centered i.i.d. r.v. valued in $\text{rr}$ representing the noise. When ${\theta }_n\to 1$, the moderate deviations principle for empirical covariance is discussed and as statistical applications we provide the moderate deviation estimates of the least square and the Yule-Walker estimators of the parameter ${\theta }_n$.