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Stochastic Process Leading to Catalan Number Recurrence

Author

Listed:
  • Mariusz Białecki

    (Institute of Geophysics Polish Academy of Sciences, Księcia Janusza 64, 01-452 Warsaw, Poland)

Abstract

Motivated by a simple model of earthquake statistics, a finite random discrete dynamical system is defined in order to obtain Catalan number recurrence by describing the stationary state of the system in the limit of its infinite size. Equations describing dynamics of the system, represented by partitions of a subset of { 1 , 2 , … , N } , are derived using basic combinatorics. The existence and uniqueness of a stationary state are shown using Markov chains terminology. A well-defined mean-field type approximation is used to obtain block size distribution and the consistency of the approach is verified. It is shown that this recurrence asymptotically takes the form of Catalan number recurrence for particular dynamics parameters of the system.

Suggested Citation

  • Mariusz Białecki, 2023. "Stochastic Process Leading to Catalan Number Recurrence," Mathematics, MDPI, vol. 11(24), pages 1-12, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4953-:d:1300116
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