Palm–Khintchine theorem

According to Heyman and Sobel (2003),[1] the theorem states that the superposition of a large number of independent equilibrium renewal processes, each with a finite intensity, behaves asymptotically like a Poisson process:

Let ${\displaystyle \{N_{i}(t),t\geq 0\},i=1,2,\ldots ,m}$ be independent renewal processes and ${\displaystyle \{N(t),t>0\}}$ be the superposition of these processes. Denote by ${\displaystyle X_{2jm}}$ the time between the first and the second renewal epochs in process ${\displaystyle j}$. Define ${\displaystyle N_{jm}(t)}$ the ${\displaystyle j}$th counting process, ${\displaystyle F_{jm}(t)=P(X_{2jm}\leq t)}$ and ${\displaystyle \lambda _{jm}=1/(E((X_{2jm)}))}$.

If the following assumptions hold

1) For all sufficiently large ${\displaystyle m}$: ${\displaystyle \lambda _{1m}+\lambda _{2m}+\cdots +\lambda _{mm}=\lambda <\infty }$

2) Given ${\displaystyle \varepsilon >0}$, for every ${\displaystyle t>0}$ and sufficiently large ${\displaystyle m}$: ${\displaystyle F_{jm}(t)<\varepsilon }$ for all ${\displaystyle j}$

then the superposition ${\displaystyle N_{0m}(t)=N_{1m}(t)+N_{2m}(t)+\cdots +N_{mm}(t)}$ of the counting processes approaches a Poisson process as ${\displaystyle m\to \infty }$.

• Law of large numbers