16. Some Probability and Stochastic Convergence Fundamentals

  1. Rik Pintelon and
  2. Johan Schoukens

Published Online: 13 APR 2012

DOI: 10.1002/9781118287422.ch16

System Identification: A Frequency Domain Approach, Second Edition

System Identification: A Frequency Domain Approach, Second Edition

How to Cite

Pintelon, R. and Schoukens, J. (2012) Some Probability and Stochastic Convergence Fundamentals, in System Identification: A Frequency Domain Approach, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/9781118287422.ch16

Publication History

  1. Published Online: 13 APR 2012
  2. Published Print: 19 MAR 2012

ISBN Information

Print ISBN: 9780470640371

Online ISBN: 9781118287422

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Keywords:

  • stochastic convergence;
  • covariance matrix;
  • stochastic limits;
  • asymptotic properties of estimators;
  • discrete Fourier transform

Summary

This chapter contains sections titled:

  • Notations and Definitions

  • The Covariance Matrix of a Function of a Random Variable

  • Sample Variables

  • Mixing Random Variables

  • Preliminary Example

  • Definitions of Stochastic Limits

  • Interrelations between Stochastic Limits

  • Properties of Stochastic Limits

  • Laws of Large Numbers

  • Central Limit Theorems

  • Properties of Estimators

  • Cramér-Rao Lower Bound

  • How to Prove Asymptotic Properties of Estimators?

  • Pitfalls

  • Preliminary Example—Continued

  • Properties of the Noise after a Discrete Fourier Transform

  • Exercises

  • Appendixes