Compound Poisson process

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 $\lambda > 0$ and jump size distribution ''G'', is a process $\$ given by

:$Y(t) = \sum_^ D_i$

where, $\$ is a counting of a Poisson process with rate $\lambda$, and $\$ are independent and identically distributed random variables, with distribution function ''G'', which are also independent of $\.\,$

When $D_i$ 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.

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:

:$\operatorname E(Y(t)) = \operatorname E(D_1 + \cdots + D_) = \operatorname E(N(t))\operatorname E(D_1) = \operatorname E(N(t)) \operatorname E(D) = \lambda t \operatorname E(D).$

Making similar use of the law of total variance, the variance can be calculated as:
:
\begin
\operatorname(Y(t)) &= \operatorname E(\operatorname(Y(t)\mid N(t))) + \operatorname(\operatorname E(Y(t)\mid N(t))) \\ pt&= \operatorname E(N(t)\operatorname(D)) + \operatorname(N(t) \operatorname E(D)) \\ pt&= \operatorname(D) \operatorname E(N(t)) + \operatorname E(D)^2 \operatorname(N(t)) \\ pt&= \operatorname(D)\lambda t + \operatorname E(D)^2\lambda t \\ pt&= \lambda t(\operatorname(D) + \operatorname E(D)^2) \\ pt&= \lambda t \operatorname E(D^2).
\end

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

:$\Pr(Y(t)=i) = \sum_n \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)$

:
\begin
\operatorname E(e^) & = \sum_i e^ \Pr(Y(t)=i) \\ pt& = \sum_i e^ \sum_ \Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n) \\ pt& = \sum_n \Pr(N(t)=n) \sum_i e^ \Pr(Y(t)=i\mid N(t)=n) \\ pt& = \sum_n \Pr(N(t)=n) \sum_i e^\Pr(D_1 + D_2 + \cdots + D_n=i) \\ pt& = \sum_n \Pr(N(t)=n) M_D(s)^n \\ pt& = \sum_n \Pr(N(t)=n) e^ \\ pt& = M_(\ln(M_D(s))) \\ pt& = e^.
\end

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.

:$\mu(A) = \Pr(D \in A).\,$

Let ''δ''₀ be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of ''Y''(''t'') is the measure

:$\exp(\lambda t(\mu - \delta_0))\,$

where the exponential exp(''ν'') of a finite measure ''ν'' on Borel subsets of the real line is defined by

:$\exp(\nu) = \sum_^\infty$

and

:$\nu^ = \underbrace_$

is a convolution of measures, and the series converges 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

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

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