Renewal Theory

Abstract

We now turn to processes that return repeatedly to their original state (e.g., betting $1 on the flip of a coin and returning to the state where one has neither earned nor lost money, called “breaking even”). In this chapter we discuss only a special case of the renewal process: flipping a coin repeatedly until a specific pattern (e.g., HTHTH) is generated. Once achieved (the act of actual renewal) the game is reset and restarted. To make things more general, we will allow the probability of H to have any value: instead of flipping a coin, we can roll a die, for example, creating patterns of sixes and nonsixes. Eventually, we play two such patterns against each other.