Editing Compound Poisson process (section)

{{Short description|Random process in probability theory}}
{{Refimprove|date=September 2014}}
A '''compound Poisson process''' is a continuous-time [[stochastic process]] with jumps. The jumps arrive randomly according to a [[Poisson process]] and the size of the jumps is also random, with a specified probability distribution. To be precise, a compound Poisson process, parameterised by a rate <math>\lambda > 0</math> and jump size distribution ''G'', is a process <math>\{\,Y(t) : t \geq 0 \,\}</math> given by

:<math>Y(t) = \sum_{i=1}^{N(t)} D_i</math>

where, <math> \{\,N(t) : t \geq 0\,\}</math> is the counting variable of a [[Poisson process]] with rate <math>\lambda</math>, and <math> \{\,D_i : i \geq 1\,\}</math> are independent and identically distributed random variables, with distribution function ''G'', which are also independent of <math> \{\,N(t) : t \geq 0\,\}.\,</math>

When <math> D_i </math> are non-negative integer-valued random variables, then this compound Poisson process is known as a '''stuttering Poisson process.''' {{source needed|date=December 2024}}