Explicit Bivariate Rate Functions for Large Deviations in AR(1) and MA(1) Processes with Gaussian Innovations

TL;DR

This paper derives explicit large deviation rate functions for sums and quadratic forms of AR(1) and MA(1) processes with Gaussian innovations, providing new proofs and detailed probabilistic insights.

Contribution

It provides explicit bivariate rate functions for sums and quadratic forms in AR(1) and MA(1) processes, including new proofs for known estimators' deviation functions.

Findings

Explicit rate functions for sums and quadratic forms in AR(1) and MA(1) processes.

New proof for deviation function of the Yule-Walker estimator.

Application of Contraction Principle to derive additional rate functions.

Abstract

We investigate large deviations properties for centered stationary AR(1) and MA(1) processes with independent Gaussian innovations, by giving the explicit bivariate rate functions for the sequence of random vectors $(\boldsymbol{S}_n)_{n \in \N} = \left(n^{-1}(\sum_{k=1}^n X_k, \sum_{k=1}^n X_k^2)\right)_{n \in \N}$. In the AR(1) case, we also give the explicit rate function for the bivariate random sequence $(\W_n)_{n \geq 2} = \left(n^{-1}(\sum_{k=1}^n X_k^2, \sum_{k=2}^n X_k X_{k+1})\right)_{n \geq 2}$. Via Contraction Principle, we provide explicit rate functions for the sequences $(n^{-1} \sum_{k=1}^n X_k)_{n \in \N}$, $(n^{-1} \sum_{k=1}^n X_k^2)_{n \geq 2}$ and $(n^{-1} \sum_{k=2}^n X_k X_{k+1})_{n \geq 2}$, as well. In the AR(1) case, we present a new proof for an already known result on the explicit deviation function for the Yule-Walker estimator.