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<p>We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order <i>k</i> and Poisson distribution of order infinity. An explicit representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process (CPP) and time-fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. The martingale characterization results for the CPP and the TFCPP are established. Finally, the fractional diffserential equation associated with the TFCPP is also obtained. Some possible applications to insurance are pointed out.</p>

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A Unified Approach to Compound Poisson Process and its Time-Fractional Versions

  • 期刊论文

  • 发表日期：

    2026年2月23日

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  • Palaniappan Vellaisamy,
  • Tomoyuki Ichiba

摘要

We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order k and Poisson distribution of order infinity. An explicit representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process (CPP) and time-fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. The martingale characterization results for the CPP and the TFCPP are established. Finally, the fractional diffserential equation associated with the TFCPP is also obtained. Some possible applications to insurance are pointed out.