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Hyperbolic decay time series

Hyperbolic decay time series such as, fractional Gaussian noise (FGN) or fractional autoregressive moving-average (FARMA) process, each exhibit two distinct types of behaviour: strong persistence or antipersistence. Beran (1994) characterized the family of strongly persistent time series. A more general family of hyperbolic decay time series is introduced and its basic properties are characterized in terms of the autocovariance and spectral density functions. The random shock and inverted form representations are derived. It is shown that every strongly persistent series is the dual of an antipersistent series and vice versa. The asymptotic generalized variance of hyperbolic decay time series with unit innovation variance is shown to be infinite which implies that the variance of the minimum mean-square error one-step linear predictor using the last $k$ observations decays slowly to the innovation variance as $k$ gets large.