# Compound Poisson process

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{{Short description|Random process in probability theory}}{{One source|date=March 2026}}
A '''compound Poisson process''' is a continuous-time [stochastic process](/source/stochastic_process) with jumps. The jumps arrive randomly according to a [Poisson process](/source/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](/source/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><ref>{{Cite book |last=Ross |first=Sheldon M. |title=Stochastic processes |date=1996 |publisher=Wiley |isbn=978-0-471-12062-9 |edition=2nd |series=Wiley series in probability and statistics |location=New York}}</ref>

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}}

==Properties of the compound Poisson process==
The [expected value](/source/expected_value) of a compound Poisson process can be calculated using a result known as [Wald's equation](/source/Wald's_equation) as:

:<math>\operatorname E(Y(t)) = \operatorname E(D_1 + \cdots + D_{N(t)}) = \operatorname  E(N(t))\operatorname E(D_1) = \operatorname  E(N(t)) \operatorname E(D) = \lambda t \operatorname  E(D).</math>

Making similar use of the [law of total variance](/source/law_of_total_variance), the [variance](/source/variance) can be calculated as: 
:<math>
\begin{align}
\operatorname{var}(Y(t)) &= \operatorname  E(\operatorname{var}(Y(t)\mid N(t))) + \operatorname{var}(\operatorname E(Y(t)\mid N(t))) \\[5pt]
&= \operatorname E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t) \operatorname E(D)) \\[5pt]
&= \operatorname{var}(D) \operatorname E(N(t)) + \operatorname E(D)^2 \operatorname{var}(N(t)) \\[5pt]
&= \operatorname{var}(D)\lambda t + \operatorname E(D)^2\lambda t \\[5pt]
&= \lambda t(\operatorname{var}(D) + \operatorname E(D)^2) \\[5pt]
&= \lambda t \operatorname E(D^2).
\end{align}
</math>

Lastly, using the [law of total probability](/source/law_of_total_probability), the [moment generating function](/source/moment_generating_function) can be given as follows:
 
:<math>\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) </math>

:<math>
\begin{align}
\operatorname E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\[5pt]
& = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) \\[5pt]
& = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i\mid N(t)=n) \\[5pt]
& = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\[5pt]
& = \sum_n \Pr(N(t)=n) M_D(s)^n \\[5pt]
& = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\[5pt]
& = M_{N(t)}(\ln(M_D(s))) \\[5pt]
& = e^{\lambda t \left( M_D(s) - 1 \right) }.
\end{align}
</math>

==Exponentiation of measures==
Let ''N'', ''Y'', and ''D'' be as above.  Let ''μ'' be the probability measure according to which ''D'' is distributed, i.e.

:<math>\mu(A) = \Pr(D \in A).\,</math>

Let ''δ''<sub>0</sub> be the trivial probability distribution putting all of the mass at zero.  Then the [probability distribution](/source/probability_distribution) of ''Y''(''t'') is the measure

:<math>\exp(\lambda t(\mu - \delta_0))\,</math>

where the exponential exp(''ν'') of a finite measure ''ν'' on [Borel subsets](/source/Borel_set) of the [real line](/source/real_number) is defined by

:<math>\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}</math>

and

:<math> \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}</math>

is a [convolution](/source/convolution) of measures, and the series converges [weakly](/source/convergence_of_random_variables).

==See also==
* [Poisson process](/source/Poisson_process)
* [Poisson distribution](/source/Poisson_distribution)
* [Compound Poisson distribution](/source/Compound_Poisson_distribution)
* [Non-homogeneous Poisson process](/source/Non-homogeneous_Poisson_process)
* [Campbell's formula](/source/Campbell's_formula) for the [moment generating function](/source/moment_generating_function) of a compound Poisson process

== References ==
{{Reflist}}{{Stochastic processes}}

{{DEFAULTSORT:Compound Poisson Process}}
Category:Poisson point processes
Category:Lévy processes

[de:Poisson-Prozess#Zusammengesetzte Poisson-Prozesse](/source/de%3APoisson-Prozess)

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Adapted from the Wikipedia article [Compound Poisson process](https://en.wikipedia.org/wiki/Compound_Poisson_process) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Compound_Poisson_process?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.