Lundberg approximations for compound distributions with insurance applications

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This monograph explores Lundberg approximations for compound distributions, focusing on their applications in insurance risk modeling. While these distributions can be analytically challenging, they are crucial in insurance and applied probability fields like queueing theory, making the content relevant for researchers and graduate students. The material is self-contained, though familiarity with insurance risk theory is advantageous for readers. Lundberg asymptotics and bounds have historically been linked to ruin probabilities and waiting time distributions, and their recent extensions to compound distributions stem from the compound geometric representation of these probabilities. The book systematically addresses these approximations, utilizing monotonicity concepts from reliability theory. Applications include solving defective renewal equations, analyzing the timing and severity of insurance ruin, and examining renewal risk models, which can also relate to the equilibrium waiting time distribution in the G/G/1 queue. Many established results are derived and expanded upon, with much of the content being novel. A distinctive aspect is the reliance on elementary analytic techniques, requiring only undergraduate-level mathematics. New proofs are presented, accompanied by an extensive bibliography. Gordon Willmot, a Professor of Statistics and Actuarial Science at the University of Waterloo, specializes in insurance risk and