Almost-s sure summability of Poisson-arrival double sequences

Let $f\in L^2_{\operatorname{imp}}$ , let $(A_k)$ be pairwise disjoint Borel subsets of $\mathbb R_+$ , and define $f_k=f\mathbf 1_{A_k}$ . Let $(S_n)$ be the arrival times of a Poisson process. Double-series summability conjecture. The double sequence $f_k(S_n)$ is summable almost surely. This concerns the interchange and joint convergence of the decomposition by disjoint spatial sets and the Poisson-arrival index; the preceding finite-partition argument gives a related reordered series but does not settle the original infinite double sequence.

References

Primary source

Jerzy Szulga, “Improper Poisson integral of the sinc function”, arXiv:2504.05574 (2025).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader has proposed a complete counterexample using the sinc function, but it has not been independently verified.

The conjecture asks whether the disjoint-set decomposition of Poisson arrivals is almost surely unconditionally summable. Szulga's 2025 preprint studies improper Poisson integrals for the sinc function and leaves the general case to conjectures.

Posted attempt

An attempted complete disproof uses $f(x)=\sin(x)/x$ and the partition $A_k=[k-1,k)$ . It claims $\sum_n |f(S_n)|=\infty$ almost surely, while the arrival-ordered series converges, so the asserted unconditional double summability fails. The attempt has not been independently verified.

Current status (as of August 2026): a complete counterexample has been claimed, but the conjecture is not verified as false and no independently confirmed resolution was found.

Sources

arxiv.org

CounterexampleThis solution needs a summarySee full solutionHide full solution

The conjecture is false under the source's explicit definition of “summable” as unconditionally summable.

Let $N$ be a rate-one Poisson process with arrival times $S_1<S_2<\cdots$ . Take

f(x)= \begin{cases} \sin(x)/x,&x>0,\\ 1,&x=0, \end{cases} \qquad A_k=[k-1,k)\quad(k\ge1).

The sets $A_k$ form a disjoint Borel partition of $[0,\infty)$ . Moreover,

f\in L^\infty\cap L^2, \qquad \int_0^\infty f(x)\,dx=\frac{\pi}{2}

as an improper integral, so $f$ belongs to the required impaired-integrability space.

Nevertheless,

\int_0^\infty\frac{|\sin x|}{x}\,dx=\infty.

Indeed, on

J_m=[m\pi+\pi/6,\ m\pi+5\pi/6]

we have $|\sin x|\ge1/2$ , giving

\int_{J_m}\frac{|\sin x|}{x}\,dx \ge\frac1{3(m+1)},

and the corresponding harmonic series diverges.

Since $|f|\le1$ , we also have

1-e^{-|f(x)|}\ge(1-e^{-1})|f(x)|.

The Poisson Laplace functional therefore yields

\mathbb E\exp\left(-\sum_{n\ge1}|f(S_n)|\right) = \exp\left(-\int_0^\infty(1-e^{-|f(x)|})\,dx\right) =0.

Consequently

\sum_{n\ge1}|f(S_n)|=\infty \qquad\text{almost surely}.

Because every arrival belongs to exactly one $A_k$ , almost surely

\sum_{k\ge1}\sum_{n\ge1} \left|f(S_n)\mathbf1_{A_k}(S_n)\right| = \sum_{n\ge1}|f(S_n)| =\infty.

Thus the asserted double family is not unconditionally summable almost surely.

The naturally arrival-ordered series still converges almost surely and in $L^2$ : it is the sum of an $L^2$ -bounded compensated Poisson martingale and the convergent improper deterministic integral. The failure concerns exactly the stronger unconditional double-summability claim.