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
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
: ![{\displaystyle \lambda _{1m}+\lambda _{m}+\cdots +\lambda _{mm}=\lambda <\infty }](//wikimedia.org/api/rest_v1/media/math/render/svg/51da89d2e73287019830b199e5ac37ddab6a63c4)
2) Given
, for every
and sufficiently large
:
for all ![{\displaystyle j}](//wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0)
then the superposition
of the counting processes approaches a Poisson process as
.