An extension of Lai and Wei's law of the iterated logarithm with applications to time series analysis and regression

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For a double array a N = ( a Nn ) −∞<n<∞ of nonrandom constants and independent random variables ε n , under some conditions, Lai and Wei (1982, Ann. Statist. 10 320–335) proved the law of the iterated logarithms for S N = Σ n a Nn ε n . This paper extends this law to the case of S ̃ N = Σ na Nnu n , where { u n } is a linear series of the form u n = Σ j=0 ∞ κ j ε n− j . As applications of the theorem we establish the law of the iterated logarithm for finite Fourier transforms in spectral analysis and for least squares estimates in regression models in which the random disturbances form a linear series.