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<p>In this paper, by using a Weyl-type operator, the notion of the <i>n</i>-th order equilibrium function of a given function is introduced and higher-order equilibrium properties for the solution of a defective renewal equation are studied. It is shown that the <i>n</i>-th order equilibrium of such solution also satisfies a defective renewal equation. Furthermore, convolution representations for these functions are given. Several applications for compound geometric distributions and for convolutions involving a compound geometric distribution are studied. Further expressions for functions with interest in ruin theory are obtained, as well as mixture representations. Finally, some bounds and applications are also provided to the classical risk model.</p>

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On Defective Renewal Equations and Compound Geometric Distributions, with Applications in Ruin Theory

  • 期刊论文

  • 发表日期：

    2025年6月25日

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  • Stathis Chadjiconstantinidis,
  • Georgios Psarrakos

摘要

In this paper, by using a Weyl-type operator, the notion of the n-th order equilibrium function of a given function is introduced and higher-order equilibrium properties for the solution of a defective renewal equation are studied. It is shown that the n-th order equilibrium of such solution also satisfies a defective renewal equation. Furthermore, convolution representations for these functions are given. Several applications for compound geometric distributions and for convolutions involving a compound geometric distribution are studied. Further expressions for functions with interest in ruin theory are obtained, as well as mixture representations. Finally, some bounds and applications are also provided to the classical risk model.