Abstract
A compound Poisson process, in this context abbreviated to cPp, is defined by a probability distribution of the number m of events in the interval (o, τ) of the original scale of the process parameter, assumed to be one-dimensional, in the following form.where du shall be inserted for t, λτ being the intensity function of a Poisson process with the expected number t of events in the interval (O, τ) and U(ν, τ) is the distribution function of ν for every fixed value of τ, here called the risk distribution. If the inverse of is substituted for τ, in the right membrum of (1), the function obtained is a function of t.If the risk distribution is defined by the general form U(ν, τ) the process defined by (1) is called a cPp in the wide sense (i.w.s.). In the sequel two particular cases for U(ν, τ) shall be considered, namely when it has the form of distribution functions, which define a primary process being stationary (in the weak sense) or non-stationary, and when it is equal to U1(ν) independently of τ. The process defined by (1) is in these cases called a stationary or non-stationary (s. or n.s.)cPp and a cPpin the narrow sense (i.n.s.) respectively. If a process is non-elementary i.e. the size of one change in the random function constituting the process is a random variable, the distribution of this variable conditioned by the hypothesis that such a change has occurred at τ is here called the change distribution and denoted by V(x, τ), or, if it is independent of τ, by V1(x). In an elementary process the size of one change is a constant, so that, in this case, the change distribution reduces to the unity distribution E(x — k), where E(ξ) is equal to I, o, if ξ is non-negative, negative respectively, and k is a given constant.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.