Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^?1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.