The SLEX model of a non-stationary random process

Hernando Ombao*, Jonathan Raz, Rainer Von Sachs, Wensheng Guo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

66 Scopus citations


We propose a new model for non-stationary random processes to represent time series with a time-varying spectral structure. Our SLEX model can be considered as a discrete time-dependent Cramér spectral representation. It is based on the so-called Smooth Localized complex EXponential basis functions which are orthogonal and localized in both time and frequency domains. Our model delivers a finite sample size representation of a SLEX process having a SLEX spectrum which is piecewise constant over time segments. In addition, we embed it into a sequence of models with a limit spectrum, a smoothly in time varying "evolutionary" spectrum. Hence, we develop the SLEX model parallel to the Dahlhaus (1997, Ann. Statist., 25, 1-37) model of local stationarity, and we show that the two models are asymptotically mean square equivalent. Moreover, to define both the growing complexity of our model sequence and the regularity of the SLEX spectrum we use a wavelet expansion of the spectrum over time. Finally, we develop theory on how to estimate the spectral quantities, and we briefly discuss how to form inference based on resampling (bootstrapping) made possible by the special structure of the SLEX model which allows for simple synthesis of non-stationary processes.

Original languageEnglish (US)
Pages (from-to)171-200
Number of pages30
JournalAnnals of the Institute of Statistical Mathematics
Issue number1
StatePublished - 2002
Externally publishedYes


  • Bootstrap
  • Fourier functions
  • Haar wavelet representation
  • Locally stationary time series
  • Periodograms
  • SLEX functions
  • Spectral estimation
  • Stationary time series

ASJC Scopus subject areas

  • Statistics and Probability


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