Abstract
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.
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References
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Kasteleyn, P.W. Variations on a theme by Mark Kac. J Stat Phys 46, 811–827 (1987). https://doi.org/10.1007/BF01011143
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DOI: https://doi.org/10.1007/BF01011143