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 *Z*_{0} corresponds to the spacing containing the origin, then the *Z*_{n} are independent, with all but *Z*_{0} being of unit Exponential distribution and *Z*_{0 }being of Exponential distribution of mean 2. (The discrepant behaviour of the *Z*_{0 }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 *i*_{1} < i_{2} < ... < 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 *i*_{1} < *i*_{2} < ... < *i*_{k} and *j*_{1} < *j*_{2} < ... <* **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 *i*_{1} < *i*_{2} < ... < *i*_{k}.

End of proof.

## Notes:

- 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.

- So is the notion of a Poisson (point) process.
- 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.
- The result generalizes easily to stationary renewal processes for which the inter-point spacing has a probability density.
- 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 ...