Typical Assumptions

  • Jan Beran


In general, consistent estimation of P is not possible without additional assumptions. In this chapter, typical assumptions used in time series analysis are discussed. For simplicity we focus on equidistant univariate real valued time series \(X_{t}\in \mathbb {R}\) (\(t\in \mathbb {Z}\)).


  1. Albiac, F., & Kalton, N. J. (2016). Topics in Banach space theory. New York: Springer.Google Scholar
  2. Beran, J. (1994). Statistics for long-memory processes. New York: Chapman & Hall/CRC Press.Google Scholar
  3. Beran, J., Feng, Y., Ghosh, S., & Kulik, R. (2013). Long-memory processes. New York: Springer.Google Scholar
  4. Birkhoff, G. D. (1931). Proof of the ergodic theorem. Proceedings of the National Academy of Sciences USA, 17, 656–660.Google Scholar
  5. Bradley, R. C. (2007). Introduction to strong mixing conditions (Vols. 1, 2, and 3). Heber City, UT: Kendrick Press.Google Scholar
  6. Breiman, L. (1992). Probability. Philadelphia: SIAM.Google Scholar
  7. Doob, J. L. (1953). Stochastic processes. New York: Wiley.Google Scholar
  8. Doukhan, P. (1995). Mixing: Properties and examples. New York: Springer-Verlag.Google Scholar
  9. Giraitis, L., Koul, H. L., & Surgailis, D. (2012). Large sample inference for long memory processes. London: Imperial College Press.Google Scholar
  10. Kallenberg, O. (2002). Foundations of modern probability. New York: Springer.Google Scholar
  11. McKean, H. (2014). Probability: The classical limit theorems. Cambridge: Cambridge University Press.Google Scholar
  12. Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proceedings of the National Academy of Sciences USA, 42, 43–47.Google Scholar
  13. Rynne, B. P., & Youngson, M. A. (2007). Linear functional analysis. London: Springer.Google Scholar
  14. Stout, H. F. (1974). Almost sure convergence. New York: Academic Press.Google Scholar
  15. Tong, H. (1990). Non-linear time series: A dynamical system approach. Oxford: Oxford University Press.Google Scholar
  16. Walters, P. (1982). An introduction to ergodic theory. New York: Springer.Google Scholar
  17. Young, N. (1988). An introduction to Hilbert space. Cambridge: Cambridge University Press.Google Scholar

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Authors and Affiliations

  • Jan Beran
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of KonstanzKonstanzGermany

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