A log log law for unstable ARMA models with applications to time series analysis

Based on a martingale analogue of Kolmogorov's law of the iterated logarithm, we obtained a log log law for unstable ARMA processes, that is, , a.s., and (, a.s., where b is an arbitrary constant, , {X(k)} is an unstable ARMA process [phi](B) X(n) = C(B) [var epsilon](n), d is the largest multiplicity of all the distinct roots of [phi](z) on the unit circle, and a = 2d - 1. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters and on iterated logarithm results for autocorrelations of unstable ARMA models.