A compound Poisson process is a real-valued point process {Zt , t ≥ 0} having the following properties. 1. finite jumps: for all ω ∈Ω, t 7−→ Zt(ω) has finitely many jumps in finite intervals. 2. independent increments: for all t,s≥ 0;Zt+s−Zt is independent of past {Zu,u≤ t}. 3. stationary increments: for all t,s≥ 0, distribution of Zt+s−Zt depends only on s and not on t. For each ω ∈Ω, we can d...

One of two simple hypotheses is correct about the unknown arrival rate and jump distribution of a compound Poisson process. We start observing the process, and the problem is to decide on the correct hypothesis as soon as possible and with the smallest probability of wrong decision. We find a Bayes-optimal sequential decision rule and describe completely how to calculate its parameters without ...

By dealing with interarrival times as exponentially distributed fuzzy random variables, a fuzzy random homogeneous Poisson process and a fuzzy random compound Poisson process are respectively defined. Several theorems on the two processes are provided, respectively.

Two cumulative damage models are considered, the inverse gamma process and a composed gamma process. They can be seen as ‘continuous’ analogues of Poisson and compound Poisson processes, respectively. For these models the first passage time distribution functions are derived. Inhomogeneous versions of these processes lead to models closely related to the Weibull failure model. All models show i...