NEGATIVE BINOMIAL SUMS OF RANDOM-VARIABLES AND DISCOUNTED REWARD PROCESSES

Given a sequence of random variables (rewards), the Haviv-Puterman dif ferential equation relates the expected infinite-horizon lambda-discou nted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with pa rameters 2 and lambda. This paper provides an interpretation of this p roven, but previously unexplained, result. Furthermore, the interpreta tion is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time deter mined by an independent negative binomial random variable with paramet ers k and lambda.