Recurrent Sequences Play for Survival Probability of Discrete Time Risk Model

In this article we investigate a homogeneous discrete time risk model with a generalized premium income rate which can be any natural number. We derive theorems and give numerical examples for finite and ultimate time survival probability calculation for the mentioned model. Our proved statements for ultimate time survival probability calculation, at some level, are similar to the previously known statements for non-homogeneous risk models, where required initial values of survival probability for some recurrent formulas are gathered by certain limit laws. We also give a simplified proof that a ruin is almost unavoidable with a neutral net profit condition and state several conjectures on a certain type of recurrent matrices non-singularity. All the research done can be interpreted as a possibility that symmetric or asymmetric random walk (r.w.) hits (or not) the line u+kappa t and that possibility is directly related to the expected value of r.w. generating random variable which might be equal, above or bellow kappa.