Contraction Properties

  • Philippe Bougerol
  • Jean Lacroix
Part of the Progress in Probability and Statistics book series (PRPR, volume 8)


We now consider i.i.d. invertible random matrices Y1, Y2, ... of arbitrary order, say d . This chapter is devoted to the study of the basic almost sure properties of the products Sn = Yn...Y1. We shall derive their salient feature, which is the contracting action of Sn on the set of directions. In particular we shall give (see 3.4, 4.3 and 6.1), following Guivarc’h and Raugi [34], a condition ensuring that
  1. (a)
    For any x ≠ 0 in ℝd, a.s. (
    $$ \mathop{{\lim }}\limits_{{n \to \infty }} \frac{1}{n}\;Log\;\left\| {{S_{n}}x} \right\| = \gamma $$
  2. (b)

    There exists a unique invariant distribution on P(ℝd).

  3. (c)
    For any x̄, ȳ in P(ℝd) , a.s.
    $$ \mathop{{{\mathop{\rm li}\nolimits} \bar{m}}}\limits_{{n \to \infty }} \frac{1}{n}\;Log\;\delta \left( {{S_{n}}\bullet \bar{x},S{}_{n}\bullet \bar{y}} \right)\quad < 0 $$
    where δ is the natural angular metric on P(ℝd).
  4. (d)

    The two upper Lyapunov exponents associated with (Sn) are distinct.



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Copyright information

© Birkhäuser Boston, Inc. 1985

Authors and Affiliations

  • Philippe Bougerol
    • 1
  • Jean Lacroix
    • 2
  1. 1.UER de MathématiquesUniversité Paris 7ParisFrance
  2. 2.Département de MathématiquesUniversité de Paris XIIIVilletaneuseFrance

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