Response and Uncertainty of the Parabolic Variance PVAR to Non-Integer Exponents of the Power Law

The oscillator fluctuations are described as the phase or frequency noise spectrum, or in terms of a wavelet variance as a function of the measurement time. The spectrum is generally approximated with the 'power law', i.e., a Laurent polynomial with integer exponents of the frequency. This article provides (i) the analytical expression of the response of the wavelet variance PVAR (Parabolic Variance) to the frequency-noise spectrum in the general case of non-integer exponents of the frequency, and (ii) a useful approximate expression of the statistical uncertainty. In turn, PVAR is relevant in that it replaces the widely used modified Allan variance MVAR, featuring the identification of noise processes with fewer data.