The Moments of the Generalized Negative Binomial Distribution and Recurrence Time in Discrete Time Markov Chains
2022, v.28, Issue 1, 53-86
The classical results of the negative binomial probability distribution - with an explicit focus on the moments - are generalized by means of Homogeneous Discrete Time Markov Chains to series of stochastically independent random trials. These do not only yield two possible outcomes but two groups of them - different kinds of successes and failures with occurrence probabilities depending on the outcome of the previous trial. This generalization allows a uniform view of occupation time, $k$-th passage and recurrence time not only to a single state but also to a subset of the index set. Also, it turns out that phase type distributions and phase type renewal processes can be viewed as special cases of this concept. Our results are derived and presented consequently in matrix form, the probabilities as well as the moments. They can be applied to all Discrete Time Markov Chains, especially in computer capacity planning, performability, economics and actuarial mathematics.
Keywords: Discrete Time Markov Chains, negative binomial distribution, recurrence time, first passage time, Stirling numbers, moments, phase type distribution, phase type renewal times