en>Steelpillow: Undid revision 614519301 by 208.50.124.65 (talk) The Lemone hexagon has one non-complex solution

2014-06-26T16:30:16Z

Undid revision 614519301 by 208.50.124.65 (talk) The Lemone hexagon has one non-complex solution

en>Decora: /* See also */

2012-01-29T18:55:01Z

See also

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{{Unreferenced|date=December 2009}}
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. 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 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 variable, then this compound Poisson process is named stuttering Poisson process which has the feature that two or more events occur in a very short time .

==Properties of the compound Poisson process==
Using [[law of total expectation|conditional expectation]], the [[expected value]] of a compound Poisson process can be calculated using a result known as [[Wald's equation]] as:

:<math>\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \lambda t 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)) &= E(\operatorname{var}(Y(t)|N(t))) + \operatorname{var}(E(Y(t)|N(t))) \\
&= E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t)E(D)) \\
&= \operatorname{var}(D)E(N(t)) + E(D)^2 \operatorname{var}(N(t)) \\
&= \operatorname{var}(D)\lambda t + E(D)^2\lambda t \\
&= \lambda t(\operatorname{var}(D) + E(D)^2) \\
&= \lambda t 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|N(t)=n)\Pr(N(t)=n) </math>

:<math>
\begin{align}
E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\
& = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i|N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\
& = \sum_n \Pr(N(t)=n) M_D(s)^n \\
& = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\
& = M_{N(t)}(\ln(M_D(s)) \\
& = 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]].

==Fitting a compound Poisson process==

The parameters for independent observations of a compound Poisson process can be chosen using a [[maximum likelihood estimator]] using Simar's algorithm,<ref>{{cite doi|10.1214/aos/1176343651}}</ref> which has been shown to converge.<ref>{{cite doi|10.1214/aos/1176345890}}</ref>

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

==References==

{{reflist}}

{{Stochastic processes}}

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

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

Complex polygon - Revision history

en>Steelpillow: Undid revision 614519301 by 208.50.124.65 (talk) The Lemone hexagon has one non-complex solution

en>Decora: /* See also */