## Abstract

Time-averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied. Such a process, called a Lévy walk, describes dynamical behaviors of many physical systems, fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example. When the mean sojourn time diverges the process is non-ergodic. In that case, the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function, instead it remains random even in the limit of long measurement time. Several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges, and favorably compared to Monte Carlo simulations. Nonergodicity of the power spectrum of the process is briefly discussed, and a nonstationary Wiener-Khintchine theorem, relating the correlation functions and the power spectrum is presented. The considered situation is in full contrast to the usual assumptions of ergodicity and stationarity.

Original language | English |
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Pages (from-to) | 137-167 |

Number of pages | 31 |

Journal | Journal of Statistical Physics |

Volume | 122 |

Issue number | 1 |

DOIs | |

State | Published - Jan 2006 |

### Bibliographical note

Funding Information:This work was supported by National Science Foundation award CHE-0344930. EB is supported in part by the Israel Science Foundation. EB also thanks Center for Complexity Science, Israel.

### Funding

This work was supported by National Science Foundation award CHE-0344930. EB is supported in part by the Israel Science Foundation. EB also thanks Center for Complexity Science, Israel.

Funders | Funder number |
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National Science Foundation | CHE-0344930 |

Israel Science Foundation |

## Keywords

- Correlation function
- Lévy statistics
- Nonergodicity
- Power law
- Time series