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

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 known versions of this problem in which the number of summed extreme order statistics is fixed, here, the specified threshold can be exceeded by an unpredictable number of sample elements. In the paper, in terms of the distribution function of a separate summand, the explicit form of the distribution of the ratio of the sum of sample elements exceeding a threshold to the total sum of sample elements is formally presented. The asymptotic and limit distributions are heuristically deduced for this ratio. These distributions are convenient for practical computations. The cases are considered in which the distributions of the summands have light tails (the second moments are finite) as well as the cases in which these distributions have heavy tails (belong to the domain of attraction of a stable law). In all cases, the normalization of the ratio is described that provides the existence of a nondegenerate limit (as the number of summands infinitely increases) distribution as well as the limit distribution itself (normal for the case of light tails and stable for the case of heavy tails).