Compound Poisson process

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Compound Poisson process" – news · newspapers · books · scholar · JSTOR (September 2014) (Learn how and when to remove this message)

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

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

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

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