Decomposition of random variables with bounded hazard rates

In this paper, we derive a decomposition formula for the distribution functions of random variables. We use the decomposition formula to provide an alternative proof to the fact that a random variable whose distribution function has bounded hazard rates can be decomposed into a cascade of a finite or an infinite number of exponential stages; the random variable terminates at the end of each stage with a probability depending on the elapsed time. The decomposition formula provides an alternative explanation to the random variable generation method introduced in Shanthikumar (Operations Research, 34, 573-580, 1986) and can be used to derive the first and second moments of the number of the stages needed in the method. Based on the decomposition, the concept of 'discretized state' is introduced. The results can be viewed as an extension of the Coxian distribution and can be used to explain the uniformization (or the thinning) of point processes.