Compound Poisson process (original) (raw)

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Random process in probability theory

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 λ > 0 {\displaystyle \lambda >0} {\displaystyle \lambda >0} and jump size distribution G, is a process { Y ( t ) : t ≥ 0 } {\displaystyle \{\,Y(t):t\geq 0\,\}} {\displaystyle \{\,Y(t):t\geq 0\,\}} given by

Y ( t ) = ∑ i = 1 N ( t ) D i {\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}} {\displaystyle Y(t)=\sum _{i=1}^{N(t)}D_{i}}

where, { N ( t ) : t ≥ 0 } {\displaystyle \{\,N(t):t\geq 0\,\}} {\displaystyle \{\,N(t):t\geq 0\,\}} is the counting variable of a Poisson process with rate λ {\displaystyle \lambda } {\displaystyle \lambda }, and { D i : i ≥ 1 } {\displaystyle \{\,D_{i}:i\geq 1\,\}} {\displaystyle \{\,D_{i}:i\geq 1\,\}} are independent and identically distributed random variables, with distribution function G, which are also independent of { N ( t ) : t ≥ 0 } . {\displaystyle \{\,N(t):t\geq 0\,\}.\,} {\displaystyle \{\,N(t):t\geq 0\,\}.\,}

When D i {\displaystyle D_{i}} {\displaystyle D_{i}} are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process. [_citation needed_]

Properties of the compound Poisson process

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The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

E ⁡ ( Y ( t ) ) = E ⁡ ( D 1 + ⋯ + D N ( t ) ) = E ⁡ ( N ( t ) ) E ⁡ ( D 1 ) = E ⁡ ( N ( t ) ) E ⁡ ( D ) = λ t E ⁡ ( D ) . {\displaystyle \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).} {\displaystyle \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).}

Making similar use of the law of total variance, the variance can be calculated as:

var ⁡ ( Y ( t ) ) = E ⁡ ( var ⁡ ( Y ( t ) ∣ N ( t ) ) ) + var ⁡ ( E ⁡ ( Y ( t ) ∣ N ( t ) ) ) = E ⁡ ( N ( t ) var ⁡ ( D ) ) + var ⁡ ( N ( t ) E ⁡ ( D ) ) = var ⁡ ( D ) E ⁡ ( N ( t ) ) + E ⁡ ( D ) 2 var ⁡ ( N ( t ) ) = var ⁡ ( D ) λ t + E ⁡ ( D ) 2 λ t = λ t ( var ⁡ ( D ) + E ⁡ ( D ) 2 ) = λ t E ⁡ ( D 2 ) . {\displaystyle {\begin{aligned}\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{aligned}}} {\displaystyle {\begin{aligned}\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{aligned}}}

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

Pr ( Y ( t ) = i ) = ∑ n Pr ( Y ( t ) = i ∣ N ( t ) = n ) Pr ( N ( t ) = n ) {\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)} {\displaystyle \Pr(Y(t)=i)=\sum _{n}\Pr(Y(t)=i\mid N(t)=n)\Pr(N(t)=n)}

E ⁡ ( e s Y ) = ∑ i e s i Pr ( Y ( t ) = i ) = ∑ i e s i ∑ n Pr ( Y ( t ) = i ∣ N ( t ) = n ) Pr ( N ( t ) = n ) = ∑ n Pr ( N ( t ) = n ) ∑ i e s i Pr ( Y ( t ) = i ∣ N ( t ) = n ) = ∑ n Pr ( N ( t ) = n ) ∑ i e s i Pr ( D 1 + D 2 + ⋯ + D n = i ) = ∑ n Pr ( N ( t ) = n ) M D ( s ) n = ∑ n Pr ( N ( t ) = n ) e n ln ⁡ ( M D ( s ) ) = M N ( t ) ( ln ⁡ ( M D ( s ) ) ) = e λ t ( M D ( s ) − 1 ) . {\displaystyle {\begin{aligned}\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{aligned}}} {\displaystyle {\begin{aligned}\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{aligned}}}

Exponentiation of measures

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

μ ( A ) = Pr ( D ∈ A ) . {\displaystyle \mu (A)=\Pr(D\in A).\,} {\displaystyle \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 ⁡ ( λ t ( μ − δ 0 ) ) {\displaystyle \exp(\lambda t(\mu -\delta _{0}))\,} {\displaystyle \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 ⁡ ( ν ) = ∑ n = 0 ∞ ν ∗ n n ! {\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}} {\displaystyle \exp(\nu )=\sum _{n=0}^{\infty }{\nu ^{*n} \over n!}}

and

ν ∗ n = ν ∗ ⋯ ∗ ν ⏟ n factors {\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}} {\displaystyle \nu ^{*n}=\underbrace {\nu *\cdots *\nu } _{n{\text{ factors}}}}

is a convolution of measures, and the series converges weakly.