Compound Poisson process: Difference between revisions

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{{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 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.''' which{{source hasneeded|date=December the feature that two or more events occur in a very short time .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}D_1 + ...\cdots + D_{N_{N(t})}) = \operatorname E(N(t))\operatorname E(D_{1}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>

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Lastly, using the [[law of total probability]], the [[moment generating function]] can be given as follows:

:<math>\,\Pr(Y(t)=i) = \sum_{n}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>

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* [[Compound Poisson distribution]]

* [[Non-homogeneous Poisson process]]

* [[Fractional Poisson process]]

* [[Campbell's formula]] for the [[moment generating function]] of a compound Poisson process