References & Further Reading

References

  1. A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, 4th ed., 2002

    Chapters 9--12 cover stochastic processes, stationarity, correlation functions, and ergodicity. Our notation and development follow Papoulis closely.

  2. S. M. Ross, Introduction to Probability Models, Academic Press, 11th ed., 2014

    A more accessible treatment of stochastic processes with emphasis on applications.

  3. P. Billingsley, Probability and Measure, Wiley, 3rd ed., 1995

    The Kolmogorov extension theorem (Ch. 36) and Birkhoff ergodic theorem (Ch. 24) are treated rigorously here.

  4. G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, Oxford University Press, 3rd ed., 2001

    Chapters 8--9 provide a clear treatment of stochastic processes and stationarity at a level slightly more advanced than Papoulis.

  5. H. Stark and J. W. Woods, Probability and Random Processes with Applications to Signal Processing, Prentice Hall, 3rd ed., 2002

    Engineering-oriented treatment with many worked examples in signal processing.

  6. R. M. Gray, Probability, Random Processes, and Ergodic Properties, Springer, 2nd ed., 2009

    Rigorous treatment of ergodic theory for engineers. Chapters 3--4 cover stationarity and Toeplitz structure.

  7. S. Haykin, Adaptive Filter Theory, Prentice Hall, 4th ed., 2001

    Chapter 2 provides an excellent engineering-oriented review of WSS processes, autocorrelation, and their role in filter design.

  8. A. Leon-Garcia, Probability, Statistics, and Random Processes for Electrical Engineering, Prentice Hall, 3rd ed., 2008

    Undergraduate-level treatment with many worked examples.

  9. P. Z. Peebles Jr., Probability, Random Variables, and Random Signal Principles, McGraw-Hill, 4th ed., 2001

    Concise treatment of stochastic processes for ECE students.

  10. T. M. Cover and J. A. Thomas, Elements of Information Theory, Wiley, 2nd ed., 2006

    Section 4.4 discusses stationarity and ergodicity in the context of entropy rates for stationary processes.

  11. G. Caire and S. Shamai (Shitz), On the Capacity of Some Channels with Channel State Information, 1999

    Unified framework for ergodic and outage capacity of fading channels with CSIT.

Further Reading

For readers who want to deepen their understanding of stochastic processes and their applications.

  • Measure-theoretic foundations of stochastic processes

    P. Billingsley, 'Probability and Measure,' 3rd ed., Wiley, 1995, Ch. 36--37

    For readers heading toward information theory research, the measure-theoretic construction of stochastic processes via the Kolmogorov extension theorem is essential.

  • Ergodic theory for engineers

    R. M. Gray, 'Probability, Random Processes, and Ergodic Properties,' 2nd ed., Springer, 2009, Ch. 6--8

    Gray provides the most accessible rigorous treatment of ergodic theory aimed at engineers, connecting the abstract theory to practical estimation and coding.

  • Stochastic processes in wireless communications

    A. Goldsmith, 'Wireless Communications,' Cambridge University Press, 2005, Ch. 3--4

    Applies the stochastic process framework directly to wireless channel modeling, including fading, coherence time, and the WSSUS channel model.

  • Random matrix theory and large-dimensional stochastic processes

    A. M. Tulino and S. Verdú, 'Random Matrix Theory and Wireless Communications,' Found. Trends Commun. Inform. Theory, vol. 1, no. 1, pp. 1--182, 2004

    When multiple antennas or users are involved, the relevant stochastic process is matrix-valued. This monograph connects random matrix theory to MIMO communications — building on the scalar theory of this chapter.

ExercisesCh 14 →