Hyperbolic decay time series

TL;DR

This paper introduces a broader class of hyperbolic decay time series, characterizes their properties, and explores their duality, persistence, and antipersistence behaviors, highlighting their slow variance decay and implications for prediction.

Contribution

It generalizes the family of hyperbolic decay time series beyond previously known models and analyzes their autocovariance, spectral density, and duality properties.

Findings

Strong persistence series are dual to antipersistent series.

The asymptotic variance of the predictor is infinite for these series.

Variance of the one-step predictor decays slowly to the innovation variance.

Abstract

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.