Studies in Astronomical Time-series Analysis. VII. An Enquiry Concerning Nonlinearity, the rms–Mean Flux Relation, and Lognormal Flux Distributions

Abstract

Abstract A broad and widely used class of stationary, linear, additive time-series models can have statistical properties that many authors have asserted imply that the underlying process must be nonlinear, nonstationary, multiplicative, or inconsistent with shot noise. This result is demonstrated with exact and numerical evaluation of the model flux distribution function and dependence of flux standard deviation on mean flux (here and in the literature called the rms–flux relation). These models can (1) exhibit normal, lognormal, or other flux distributions; (2) show linear or slightly nonlinear rms–mean flux dependencies; and (3) match arbitrary second-order statistics of the time-series data. Accordingly, the above assertions cannot be made on the basis of statistical time-series analysis alone. Also discussed are ambiguities in the meaning of terms relevant to this study—linear, stationary, and multiplicative—and functions that can transform observed fluxes to a normal distribution as well as or better than the logarithm.