On first passage time structure of random walks
For continuous time birth-death processes on \({\mathbb{N}}\) the first passage time \(\tau_{n,n+1}\) from n to \(n+1\) is always a mixture of \(n+1\) exponential random variables. Furthermore the first passage time \(\tau_{0,n}\) is the sum of n independent exponential random variables. For a discrete time birth-death process the results are similar, replacing the exponential distributions by appropriate geometric ones. However if one allows the process to stay with certain probabilities in the same state, the above may not be true. The authors give necessary and sufficient conditions on the transition probabilities such that \(\tau_{n,n+1}\), \(\tau_{0,n}\) is a mixture, sum of geometric random variables.
scientific article; zbMATH DE number 3917392
scientific article