On the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements. II

The problem of description of the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements is considered. Unlike other versions of this problem in which the number of summed extreme order statistics and the threshold are fixed, here the specified threshold can be exceeded by an unpredictable number of sample elements. The situation is considered where the threshold infinitely increases as the sample size grows. It is demonstrated that in this case, the distribution of the ratio mentioned above can be approximated by the compound Poisson distribution in which the compounding law is the generalized Pareto distribution.