Generalized waiting time distributions associated with runs

Abstract

Let $\left\{ X_{t},t\ge 1\right\}$ be a sequence of random variables with two possible values as either “1” (success) or “0” (failure). Define an independent sequence of random variables $\left\{ D_{i},i\ge 1\right\}$. The random variable $D_{i}$ 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: $N_{k}$ is the number of trials until the weight of a single success run exceeds or equals k,  and $N_{r,k}$ is the number of trials until the weight of each of r success runs equals or exceeds k in $\left\{ X_{t},t\ge 1\right\}$. Distributional properties of the waiting time random variables $N_{k}$ and $N_{r,k}$ are studied and illustrative examples are presented.