Modern algebra and the Poisson point process

Here is a trivial fact about the Poisson point process, which I find curious because it links directly to a concept from modern algebra (the notion of a free abelian group).

Recall that a Poisson point process on the real line can be viewed as a (locally finite) random pattern of points. We can define it in terms of its consecutive inter-point spacing lengths, Z_n for n running through the integers. If Z₀ corresponds to the spacing containing the origin, then the Z_n are independent, with all but Z₀ being of unit Exponential distribution and Z₀ being of Exponential distribution of mean 2. (The discrepant behaviour of the Z₀ distribution is due to size-biasing, and is associated with the famous Waiting Time Paradox.)

Suppose I secretly select finitely many different spacings, Z_i1 , Z_i2 , ..., Z_ik, and tell you only the total length of the spacings I have selected, namely z = Z_i1 + Z_i2 +...+ Z_ik.

If you are allowed to inspect closely the relevant realization of the unit-rate Poisson point process, then knowledge of the total length z is all you need to determine which are the spacings i₁ < i₂ < ... < i_k that have been selected.

Indication of proof: the Z_i's are independent with Exponential distributions (unit rate except for i=0, for which the rate is 2). But this means that almost surely the spacing lengths Z_i generate a free abelian group under addition. In particular, almost surely

(Z_i1 + Z_i2 + ... + Z_ik) - (Z_j1 + Z_j2 + ... + Z_jr)

must be non-zero unless the sequences i₁ < i₂ < ... < i_k and j₁ < j₂ < ... < j_r are identical. So (since there are only countably many finite integer-coefficient linear combinations) almost surely z = Z_i1 + Z_i2 +...+ Z_ik determines the sequence i₁ < i₂ < ... < i_k.
End of proof.

Notes:

1. Free abelian groups are defined in the Encyclopaedia of Mathematics. I learnt about these in my second undergraduate year and then never thought about them again till I noticed this phenomenon just recently.
   
2. So is the notion of a Poisson (point) process.
3. For a child-friendly explanation of the Waiting Time paradox, see Masuda N and Porter MA (2021) The Waiting-Time Paradox. Frontiers for Young Minds 8:582433. DOI: 10.3389/frym.2020.582433.
4. The result generalizes easily to stationary renewal processes for which the inter-point spacing has a probability density.
5. I came across this trivial fact while working on generalizing my paper on random lines and metric spaces (Kendall, W. S. (2017). From random lines to metric spaces. Annals of Probability, 45(1), 469–517. https://doi.org/10.1214/14-AOP935); the Poisson process result is noted there (without the modern algebra adornments) in the process of proving that planar line-pattern-based geodesics between prescribed points are almost surely unique ...

Wilfrid Kendall : 12 Mar 2022 11:33 |  Tags: Poisson Point Process |  Comments (0) |  Trackbacks (1) |  Report a problem