Comments on the Epsilon and Omega cumulative distributions: “Saturation in the Hausdorff sense”

In this paper we study the asymptotic behavior of the Hausdorff distance between Heaviside function and the Epsilon and Omega distribution functions, introduced in [24], [28]. Investigations of ”saturation to the horizontal asymptote a = 1, in the Hausdorff sense” is important for professionals working in the field of financial mathematics. We prove precise bound d=lnmm+O(1m).One of the most significant goals of any insurance risk activity is to achieve a satisfactory model for the probability distribution of the total claim amount. The estimates for the value of the best Hausdorff approximation of Heaviside function by fixed cumulative distribution function give more insight on the parameters in the strategy ”Insurance responsibility”. The results are relevant for applied insurance mathematics.Numerical examples using CAS Mathematica, illustrating our results are given.