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Weakly Stationary Processes and Their Spectral Measures

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Stationary Processes and Discrete Parameter Markov Processes

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 293))

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Abstract

Stationary stochastic processes are analyzed at the level of their first and second order characteristics, mean and covariance, using Fourier methods.

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Notes

  1. 1.

    See BCPT, Chap. VI.

  2. 2.

    BCPT, p. 168.

  3. 3.

    See BCPT, p.110.

  4. 4.

    According to a celebrated Theorem 23.9 in Chapter 23, this also provides an example of a special type of associated statistical dependence.

  5. 5.

    Stationary (translation invariant) and self-similar phenomena are of rather widespread interest in the sciences, especially in connection with critical phenomena. Thus the stochastic models for such phenomena typically involve random measures and/or generalized functions.

  6. 6.

    See BCPT, p.119.

  7. 7.

    See BCPT, p. 112.

  8. 8.

    See Bhattacharya and Waymire (2021) or BCPT, p.180.

  9. 9.

    A metaphor for a stationary sequence is a process viewed as a data stream movie for which statistically it does not matter when one arrives at the theater. Time-reversibility permits it to also be run backward without changing the stochastic structure.

References

  • Bhattacharya R, Waymire E (2021) Random walk, Brownian motion, and martingales. Graduate text in mathematics. Springer, New York

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  • Brockwell P, Davis R (1991) Time series: theory and methods. Springer series in statistics. Springer, New York

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  • Grenander U (1981) Abstract inference. Wiley, New York

    MATH  Google Scholar 

  • Lévy P (1953) Random functions: general theory with special reference to Laplacian random functions (Vol. 1, No. 12). University of California Press

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

    Rabi Bhattacharya

  2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

    Edward Waymire

Authors
  1. Rabi Bhattacharya
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  2. Edward Waymire
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Bhattacharya, R., Waymire, E. (2022). Weakly Stationary Processes and Their Spectral Measures. In: Stationary Processes and Discrete Parameter Markov Processes. Graduate Texts in Mathematics, vol 293. Springer, Cham. https://doi.org/10.1007/978-3-031-00943-3_2

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