Moderate deviations for stable Markov chains and regression models. (English) Zbl 0980.62082

Worms, Julien

Electron. J. Probab. 4, Paper No. 8, 28 p. (1999).

Summary: We prove moderate deviations principles for
1) unbounded additive functionals of the form \(S_n=\sum^n_{j=1} g(X_{j-1}^{(p)}\)), where \((X_n)_{n \in\mathbb{N}}\) is a stable \(\mathbb{R}^d\)-valued functional autoregressive model of order \(p\) with white noise, and \(g\) is an \(\mathbb{R}^q\)-valued Lipschitz function of order \((r,s)\);
2) the error of the least squares estimator (LSE) of the matrix \(\theta\) in an \(\mathbb{R}^d\)-valued regression model \(X_n=\theta^t \varphi_{n-1}+ \varepsilon_n\), where \((\varepsilon_n)\) is a “generalized Gaussian” noise.
We apply these results to study the error of the LSE for a stable \(\mathbb{R}^d\)-valued linear autoregressive model of order \(p\).

MSC:

62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F10	Large deviations
60J10	Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords:

martingales; moderate deviations; least squares estimator