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# Poisson process

A **Poisson process** is a process which jumps up one. The times between jumps are independent and exponentially distributed with mean $1/\lambda$. If we put $\lambda=1$ we get a standard Poisson process. A Poisson process is an important special case of the more general class of Lévy processes.
The characteristic function of Poisson proces $N(t)$ is \[\phi_{N(t)}=E[e^{iyN(t)}]=e^{tK(y)},\] where $K(y)=iy\mu+\lambda(e^{iy}-1)$.

If we put $\mu=-\lambda$ and define $X(t)=N(t)/\sqrt{\lambda}$ then $X$ will converge in distribution to a Brownian motion as $\lambda$ tends to $\infty$.

Copyright (C) Magnus Wiktorsson 2011

### Inhomogeneous Poisson process

It is also possible to define a time-inhomogeneous Poisson process. The easist way of understanding a time-inhomogeneous Poisson process is to view it as a time-change of a standard Poisson process $N$. Let $\lambda(t)$ be a non-negative function defined on the positive real numbers (${\mathbb R}^+$). Define the cumulative intensity function $\Lambda(t)=\int_0^t\lambda(s)ds$. A time-inhomogeneous Poisson process $X$ with intensity function $\lambda(\cdot)$ can now be obtained as $X(t)=N(\Lambda(t))$.

If we put $\mu=-\lambda$ and define $X(t)=N(t)/\sqrt{\lambda}$ then $X$ will converge in distribution to a Brownian motion as $\lambda$ tends to $\infty$.

Copyright (C) Magnus Wiktorsson 2011

Questions: Magnus Wiktorsson

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