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Markov Chains pp 523-574 | Cite as

Spectral Theory

  • Randal DoucEmail author
  • Eric Moulines
  • Pierre Priouret
  • Philippe Soulier
Chapter
Part of the Springer Series in Operations Research and Financial Engineering book series (ORFE)

Abstract

Let P be a positive Markov kernel on \(\mathsf {X}\times \mathscr {X}\) admitting an invariant distribution \(\pi \). We have shown that P defines an operator on the Banach space Open image in new window . Therefore, a natural approach to the properties of P consists in studying the spectral properties of this operator. This is the main theme of this chapter, in which we first define the spectrum of P seen as an operator both on Open image in new window , \(p \ge 1\), and on an appropriately defined space of complex measures. We will also define the adjoint operator and establish some key relations between the operator norm of the operator and that of its adjoint. We also discuss geometric and exponential convergence in \(\mathrm {L}^2(\pi )\). We show that the existence of an \(\mathrm {L}^2(\pi )\)-spectral gap implies \(\mathrm {L}^2(\pi )\)-geometric ergodicity; these two notions are shown to be equivalent if the operator P is self-adjoint in \(\mathrm {L}^2(\pi )\) (or equivalently that \(\pi \) is reversible with respect to P). We extend these notions to cover Open image in new window exponential convergence in. In we introduce the notion of conductance and establish the Cheeger inequality for reversible Markov kernels.

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Randal Douc
    • 1
    Email author
  • Eric Moulines
    • 2
  • Pierre Priouret
    • 3
  • Philippe Soulier
    • 4
  1. 1.Département CITITelecom SudParisÉvryFrance
  2. 2.Centre de Mathématiques AppliquéesEcole PloytechniquePalaiseauFrance
  3. 3.Université Pierre et Marie CurieParisFrance
  4. 4.Université Paris NanterreNanterreFrance

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