## Abstract

We investigate a stochastic signal described by a renewal process for a system with N states. Each state has an associated joint distribution for the signal's intensity and its holding time. We calculate multi-point distributions, correlation functions, and the power-spectrum of the signal. Focusing on fat tailed power-law distributed sojourn times in the states of the system, we investigate 1/f noise in this widely applicable model. When the mean waiting time is infinite, the averaged sample spectrum depends both on the age of the process, i.e. the time elapsing from start of the process and the start of observation, and on the total time of observation. Fluctuations of the periodogram estimator of the power-spectrum are investigated for aged systems and are found to be determined by the distribution of the number of renewals in the observation time window. These reduce to the Mittag-Leffler distribution when the start of observation is also the start of the process. When the average waiting time is finite we find a time independent Wienerian spectrum computed from the stationary correlation function of the signal.

Original language | English |
---|---|

Pages (from-to) | 191-239 |

Number of pages | 49 |

Journal | Mathematical Modelling of Natural Phenomena |

Volume | 11 |

Issue number | 3 |

DOIs | |

State | Published - 2016 |

### Bibliographical note

Publisher Copyright:© 2016 EDP Sciences.

## Keywords

- 1/f-noise
- Ageing
- Mittag-Leffler-distribution
- Multi-point distributions
- Renewal theory
- Weak ergodicity breaking
- Wiener Khinchin theorem