Generalized Waiting Time Distributions Associated With Runs

Abstract

Let be a {X-t, t >= 1} sequence of random variables with two possible values as either "1" (success) or "0" (failure). Define an independent sequence of random variables {D-i, i >= 1}. The random variable is associated with the success when it occupies the ith place in a run of successes. We define the weight of a success run as the sum of the D values corresponding to the successes in the run. Define the following two random variables: is the number of trials until the weight of a single success run exceeds or equals k, and is the number of trials until the weight of each of r success runs equals or exceeds k in {X-t, t >= 1}. Distributional properties of the waiting time random variables and are studied and illustrative examples are presented.