Clock prediction and cross-sigma

The purpose of this paper is to describe relations between the prediction of clock time series Xm, the prediction error standard deviation σ and σ which is used to estimate σ in the presence of noise. We confine the discussion to linear predictors. These predictors are sufficient to optimally predict clocks whose stability is described by an extension of the common oscillator model, including drifts. We compare the 3-corner-hat estimate of σ of a single clock with σ and show that in the presence of noise σ is a better estimate. If all measurements are not made simultaneously, σ becomes an estimate of an auto-correlation function rather than of σ. As a result the power spectrum of the prediction errors of a single clock may be estimated