Estimation of ruin probability for random sum of negative binomially distributed number of sub-Gaussian claims

This paper investigates the properties of a classical risk process in which the number of insurance claims follows a negative binomial distribution, and the claim sizes belong to the class of strictly sub-Gaussian random variables of a generalized type. This class is broad and includes both standard sub-Gaussian and Gaussian random variables. The core assumption is that each random variable in this class satisfies a specific inequality that constrains its exponential growth, allowing for control over the behavior of its probabilistic tail. It is also assumed that the insurance premium collected over time is described by a continuous, monotone increasing function whose increments are bounded by another continuous, increasing function. This condition enables flexible modeling of premium growth over time. The aim of the paper is to obtain estimates for the probability of ruin in the described risk process model. To achieve this, the approach is based on the method of metric entropy, which allows for the consideration of both the complex nature of the claim count distribution and the properties of strictly sub-Gaussian claim sizes. This approach makes it possible to derive estimates even in cases where the exact distribution of the claims is unknown, provided that they exhibit sub-Gaussian behavior.