Abstract
Lorden’s inequality asserts that the mean return time in a renewal process with (iid) interarrival times \(Y_1, Y_2,\ldots \), is bounded above by \(2\textbf{E}[Y_1]/\textbf{E}[Y_1^2]\). We establish this result in the context of regenerative sets, and remove the factor of 2 when the regenerative set enjoys a certain monotonicity property. This property occurs precisely when the Lévy exponent of the associated subordinator is a special Bernstein function. Several equivalent stochastic monotonicity properties of such a regenerative set are demonstrated.
Dedicated to Professor Masatoshi Fukushima.
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Fitzsimmons, P.J. (2022). Monotonicity Properties of Regenerative Sets and Lorden’s Inequality. In: Chen, ZQ., Takeda, M., Uemura, T. (eds) Dirichlet Forms and Related Topics. IWDFRT 2022. Springer Proceedings in Mathematics & Statistics, vol 394. Springer, Singapore. https://doi.org/10.1007/978-981-19-4672-1_7
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