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<p>The paper studies the counting process arising as a subset of births and deaths in a birth–death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a function of the state of the birth and death process and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested. </p>

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Second order properties of thinned counts in finite birth–death processes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年2月20日

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  • Daryl. J. Daley,
  • Yoni Nazarathy,
  • Jiesen Wang

摘要

The paper studies the counting process arising as a subset of births and deaths in a birth–death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a function of the state of the birth and death process and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested.