Homogeneous Poisson Point Process

References

(Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Poisson_point_process#Homogeneous_Poisson_point_process Retrieved:2016-7-18.
- If a Poisson point process has a constant parameter, say, [math]\displaystyle{ \textstyle \lambda }[/math], then it is called a homogeneous or stationary Poisson point process. The parameter, called rate or intensity, is related to the expected (or average) number of Poisson points existing in some bounded region. In fact, the parameter [math]\displaystyle{ \textstyle \lambda }[/math] can be interpreted as the average number of points per some unit of extent such as length, area, volume, or time, depending on the underlying mathematical space, hence it is sometimes called the mean density ; see Terminology; the extent is sometimes called the exposure.^[1] ^[2]

https://stackoverflow.com/questions/31778995/how-to-generate-a-homogeneous-poisson-point-process-in-a-circle
- QUOTE: I would like to generate N points in a circle C of center (0,0) and of radius R=200. The points follow Poisson distribution. In other words, I would like to generate N homogeneous Poisson point process (HPPP) inside C. …

(Heath et al., 2013) ⇒ Robert W. Heath, Marios Kountouris, and Tianyang Bai. (2013). “Modeling Heterogeneous Network Interference Using Poisson Point Processes." IEEE Transactions on Signal Processing 61, no. 16 doi:10.1109/TSP.2013.2262679

(Pasupathy, 2010) ⇒ Raghu Pasupathy. (2010). “Generating Homogeneous Poisson Processes.” Wiley encyclopedia of operations research and management science
- ABSTRACT: We present an overview of existing methods to generate pseudorandom numbers from homogeneous Poisson processes. We provide three well-known definitions of the homogeneous Poisson process, present results that form the basis of various existing generation algorithms, and provide algorithm listings. With the intent of guiding users seeking an appropriate algorithm for a given setting, we note the computationally burdensome operations within each algorithm. Our treatment includes one-dimensional and two-dimensional homogeneous Poisson processes. Key words: statistics; simulation; random process generation; Poisson processes. Recall that a counting process {Nt, t ≥ 0} is a stochastic process defined on a sample space Ω such that for each ω ∈ Ω, the function Nt(ω) is a “realization ” of the number of “events ” happening in the interval (0, t], with N0(ω) = 0. By this definition, Nt(ω) is automatically integer valued, nondecreasing, and right-continuous for each ω. A homogeneous Poisson process is a type of counting process that is characterized as follows. Definition 1. A counting process {Nt, t ≥ 0} is called a homogeneous Poisson process if: (i) ∀t, s ≥ 0, and 0 ≤ u ≤ t, Nt+s − Nt is independent of Nu; (ii) ∀t, s ≥ 0,Pr{Nt+s − Nt ≥ 2} = o(s); and