# Deviations from normal distributions in artificial and real time series: a false positive prescription

## TL;DR

This paper investigates how deviations from normal distributions in time series can lead to false positives in identifying the underlying stochastic processes, using artificial data and real astrophysical observations.

## Contribution

It introduces a method to estimate false positive rates for PDF functional form classification in time series analysis, especially for steep PSD spectral indices.

## Key findings

- Artificial time series with steep PSDs often deviate from Gaussian PDFs.
- The false positive rate for misclassifying PDFs depends on the spectral index.
- PKS2155-304 likely has a high probability of a Gaussian PDF, given the prior.

## Abstract

Time series analysis allows for the determination of the Power Spectral Density (PSD) and Probability Density Function (PDF) for astrophysical sources. The former of these illustrates the distribution of power at various timescales, typically taking a power-law form, while the latter characterises the distribution of the underlying stochastic physical processes, with Gaussian and lognormal functional forms both physically motivated. In this paper, we use artificial time series generated using the prescription of Timmer & Koenig to investigate connections between the PDF and PSD. PDFs calculated for these artificial light curves are less likely to be well described by a Gaussian functional form for steep (<-1) PSD spectral indices due to weak non-stationarity. Using the Fermi LAT monthly light curve of the blazar PKS2155-304 as an example, we prescribe and calculate a false positive rate which indicates how likely the PDF is to be attributed an incorrect functional form. Here, we generate large numbers of artificial light curves with intrinsically normally distributed PDFs and with statistical properties consistent with observations. These are used to evaluate the probabilities that either Gaussian or lognormal functional forms better describe the PDF. We use this prescription to show that PKS2155-304 requires a high prior probability of having a normally distributed PDF, P(G) >= 0.82, for the calculated PDF to prefer a Gaussian functional form over a lognormal. We present possible choices of prior and evaluate the probability that PKS2155-304 has a lognormally distributed PDF for each.