In probability theory, the Palm–Khintchine theorem, the work of Conny Palm and Aleksandr Khinchin, expresses that a large number of not necessarily Poissonian renewal processes combined will have Poissonian properties.
According to Heyman and Sobel (2003), the theorem describes that the superposition of a large number of independent equilibrium renewal processes, each with a small intensity, behaves asymptotically like a Poisson process:
Let be independent renewal processes and be the superposition of these processes. Denote by the time between the first and the second renewal epochs in process . Define the th counting process, and .
If the following assumptions hold
1) For all sufficiently large :
2) Given , for every and sufficiently large : for all
then the superposition of the counting processes approaches a Poisson process for to .
- Daniel P. Heyman, Matthew J. Sobel (2003). "5.8 Superposition of Renewal Processes". Stochastic Models in Operations Research: Stochastic Processes and Operating Characteristics.