Editing Compound Poisson process

{{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}}

==Properties of the compound Poisson process==
The [[expected value]] of a compound Poisson process can be calculated using a result known as [[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]], the [[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]], the [[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]] 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 set|Borel subsets]] of the [[real number|real line]] 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]] of measures, and the series converges [[convergence of random variables|weakly]].

==See also==
* [[Poisson process]]
* [[Poisson distribution]]
* [[Compound Poisson distribution]]
* [[Non-homogeneous Poisson process]]
* [[Campbell's formula]] for the [[moment generating function]] of a compound Poisson process

{{Stochastic processes}}

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

[[de:Poisson-Prozess#Zusammengesetzte Poisson-Prozesse]]