Moderate deviations for stable Markov chains and regression models

Abstract

We prove moderate deviations principles for:

• 1. unbounded additive functionals of the form $S_n = \sum_{j=1}^{n} g(X^{(p)}_{j-1})$, where $(X_n)_{n\in N}$ is a stable $R^d$-valued functional autoregressive model of order $p$ with white noise and stationary distribution $\mu$, and $g$ is an $R^q$-valued Lipschitz function of order $(r,s)$;

• 2. the error of the least squares estimator (LSE) of the matrix $\theta$ in an $R^d$-valued regression model $X_n = \theta^t \phi_{n-1} + \epsilon_n$, where $(\epsilon_n)$ is a generalized gaussian noise.

We apply these results to study the error of the LSE for a stable $R^d$-valued linear autoregressive model of order $p$.