Journal of Statistical Physics

, Volume 46, Issue 5–6, pp 811–827 | Cite as

Variations on a theme by Mark Kac

  • P. W. Kasteleyn


Mark Kac's theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations, most of which are formulated in terms of 0–1 processes. Whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0, this condition is dropped in part of the variations; two others refer to the variance of the recurrence time and two to the Poincaré cycle of a dynamical system. Most variations consist in inequalities and formal identities for the mean first-arrival time and subsequent recurrence times for the given state.

Key words

Stationary stochastic process 0–1 process first-passage time recurrence time Poincaré cycle inequalities 


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Copyright information

© Plenum Publishing Corporation 1987

Authors and Affiliations

  • P. W. Kasteleyn
    • 1
  1. 1.Instituut-Lorentz voor Theoretische NatuurkundeLeidenThe Netherlands

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