The Eigenvalue Method

  • David Aldous
Part of the Applied Mathematical Sciences book series (AMS, volume 77)


Consider a stationary process (X t: t >0) and a first hitting time T = min{t: XtB}. Under many circumstances one can show that T must have an exponential tail:
$$P(T > t) \sim A\exp ( - \lambda t)ast \to \infty $$
and give an eigenvalue interpretation to λ. (In discrete time, “exponential” becomes “geometric”, of course.) In the simplest example of finite Markov chains this is a consequence of Perron-Frobenius theory reviewed below. See Seneta (1981) and Asmussen (1987) for more details.


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Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • David Aldous
    • 1
  1. 1.Department of StatisticsUniversity of California-BerkeleyBerkeleyUSA

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