Abstract
This paper investigates a curious phenomenon of some positive random variables having a constant density in some initial subinterval of their support. We discuss two methods of obtaining such a random variable with partly flat density. The main method is to sum powers of independent uniform variables with the number of terms matching the power. We show that the limit of this sum is an interesting infinitely divisible distribution and we study its basic properties, find a differential equation for its density, and obtain a recursive relation satisfied by its moments which allows for the calculation of its moment generating function and cumulants.
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