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

• Published in: Methodology and computing in applied probability Vol. 27; no. 3; p. 53
• Main Authors: Chadjiconstantinidis, Stathis, Psarrakos, Georgios
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.09.2025; Springer Nature B.V
• Subjects: Actuarial science; Aging; Business and Management; Economics; Electrical Engineering; Equilibrium; Life Sciences; Mathematical functions; Mathematics and Statistics; Operators (mathematics); Probability; Queuing theory; Random variables; Representations; Statistics; Stop-loss transform; Primary 60E05; Equilibrium distribution; Defective renewal equation; Weyl operator; Ruin theory; Residual lifetime; Aging classes; Secondary 91G05
• ISSN: 1387-5841; 1573-7713
• DOI: 10.1007/s11009-025-10178-2

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.