On the martingale approximation of the estimation error of ARMA parameters (Q807566)

The aim of this paper is to prove a theorem which is instrumental in the verifying Rissanen's tail condition for the M.L. estimation error of the parameter \(\mu^*\) of a Gaussian ARMA (p,q) process. Let \({\hat \mu}{}_ N\) (the M.L. estimator of \(\mu^*)\) be assumed to belong to an appropriate compact \(D\subset R^{p+q}\); the tail condition is that for any \(c>0:\) \[ \sum^{\infty}_{N=1}{}_{\mu^*\in D}P[N^{1/2}| {\hat \mu}_ N-\mu^*| >c \ln N]<\infty \] holds. The author presents here a significantly simplified proof of a result on the martingale approximation of the estimation parameter error. The proof is a carefull reexamination of a standard technique (linearization around the estimator \({\hat \mu}{}_ N)\) combined with inequalities-recently published by L. Gerencser - for a class of mixing processes. Many asymptotic properties of \({\hat \mu}{}_ N-\mu^*\) can be derived from those of martingales.