On Backshift-Operator polynomial transformations to stationarity for nonstationary time series and their aggregates

Abstract

It is demonstrated that any two backshift operator polynomials which transform a given nonstationary time series into weakly stationary series with continuous spectral distributions must have a common divisor which has this property. It follows that the lowest, degree polynomial with this property is unique to within a constant, multiple, Using this result some derivations are given, under varying assumptions, of a transformation formula used in nonstationary signal extraction. Counterexamples are presented to show that continuity assumptions on the spectral distribution functions involved are necessary to both obtain the uniqueness of the polynomial transformation to stationarity and also for this signal extraction formula.

Keywords:

• continous spectral distributions
• uniqueness of lowest degree polynomial
• signal extraction formula