{{Short description|Random process in probability theory}}
{{Refimprove|date=September 2014}}
A '''compound Poisson process''' is a continuous-time (random) [[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. ATo 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 athe 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.''' which has the feature that two or more events occur in a very short time.{{vaguesource needed|reasondate=How is "a very short time"December defined?2024}}
