Approximation conjecture for the ARMA point process by INARMA processes

About 8 years old · traced to 2018

Let $N$ be an ARMA point process with immigration rate $bc$ and piecewise continuous intensity $c6$ . For $c3>0$ , let $(b5_i)_{i\in\mathbb Z}$ be i.i.d. Poisson random variables with mean $bc\Delta$ , and set $\widetilde{\theta}_k=\Delta\theta(k\Delta)$ and $\widetilde{\phi}_j=\Delta\phi(j\Delta)$ . Under the summability conditions

\sum_{k=1}^{\infty}\widetilde{\theta}_k<\infty,\qquad \sum_{j=1}^{\infty}\widetilde{\phi}_j<1,

there is a stationary integer-valued time series $X^{(\Delta)}$ satisfying

X_{l}^{(\Delta)}=\epsilon_{l}+\sum_{k=1}^{\infty}\widetilde{\theta}_{k}\circ\epsilon_{l-k}+\sum_{j=1}^{\infty}\widetilde{\phi}_{j}\circ X_{l-j}^{(\Delta)},\qquad l\in\mathbb Z.

Approximation conjecture. There exists $\delta\in(0,\infty)$ such that, for every $\Delta\in(0,\delta)$ , the process above is a stationary INARMA process, and the family of point processes

N^\Delta(A)=\sum_{n\colon n\Delta\in A}X_n^\Delta

for Borel sets $A\subseteq\mathbb R$ converges weakly to $N$ as $\Delta\rightarrow0$ . Here weak convergence means vague convergence of the induced measures, equivalently convergence of finite-dimensional distributions. This conjecture formalizes the expected asymptotic equivalence between aggregated INARMA processes and the ARMA point process; the existence of the infinite-order INARMA process follows under the stated summability condition, while the asserted weak convergence as the bin width tends to zero is the substantive approximation claim.

References

Primary source

Spencer Wheatley, Michael Schatz and Didier Sornette, “The ARMA Point Process and its Estimation”, arXiv:1806.09948 (2018).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A reader-written construction claims to refute the conjecture by making the sampled process fail to exist on arbitrarily fine meshes, but it is unverified and targets a statement removed from the final paper.

Wheatley, Schatz, and Sornette formulated the conjecture in 2018: sufficiently fine discretizations should produce stationary infinite-order INARMA processes converging to the ARMA point process. Their preprint explicitly says that a rigorous proof is beyond its scope.

Known results

Under the stated summability conditions, the infinite-order INARMA process exists.
The analogous INAR-to-Hawkes weak-convergence result was already known.
Aggregated Neyman–Scott processes were reported as approximable by INMA processes.

Posted attempt

A reader-written construction chooses a smooth, integrable reproduction kernel whose point samples have divergent lattice sums for every mesh $\Delta=1/m$ , claiming that the sampled stationary process then fails to exist and thereby refuting the original universal-small-mesh assertion. It also proposes stronger tail assumptions or cell-average coefficients as repairs. The attempt has not been independently verified, and it concerns the conjecture in the withdrawn 2018 preprint; the final 2022 article reportedly removed that integer-valued-time-series discussion.

Current status (as of August 2026): The 2018 conjecture remains unproved in the primary source, while a reader-written counterexample claim gives unverified evidence against its original formulation; the status of any repaired approximation theorem remains open.

Sources

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Source-history caveat. This conjecture appears in the 2018 preprint cited by this database entry, but was removed from the final 2022 journal article: its Appendix A4 explicitly says the discussion of integer-valued time series was removed. The counterexample below addresses the withdrawn preprint statement, not an assertion retained in the published revision. It also exposes the same erroneous sampled-subcriticality assertion in a separate 2016 published lemma.

The preprint assumes a nonnegative piecewise-continuous reproduction kernel $\phi$ with

\int_0^\infty\phi(t)\,dt<1

and claims that the sampled INARMA process exists and is stationary for every sufficiently small mesh $\Delta>0$ , using reproduction coefficients

\widetilde\phi_j=\Delta\phi(j\Delta).

Choose an even smooth bump $\psi\in C_c^\infty((-1,1))$ with $0\le\psi\le1$ and $\psi(0)=1$ . For integers $n\ge2$ , set

r(n)=\lfloor\log_2n\rfloor,\qquad a_n=\frac1{2^{r(n)}(r(n)+1)}, \qquad w_n=\frac{a_n}{16},

and define

\phi(t) =\sum_{n=2}^\infty a_n\psi\left(\frac{t-n}{w_n}\right). \tag{1}

The bump supports are disjoint and locally finite. Thus $\phi$ is nonnegative, $C^\infty$ , bounded, globally Lipschitz, and tends to zero at infinity. Indeed,

\|\phi'\|_\infty \le16\|\psi'\|_\infty.

Writing $J=\int_{-1}^1\psi(u)\,du\le2$ , dyadic grouping gives

\sum_{n=2^r}^{2^{r+1}-1}a_n =\frac1{r+1}, \qquad \sum_{n=2^r}^{2^{r+1}-1}a_n^2 =\frac1{2^r(r+1)^2}.

Consequently

\int_0^\infty\phi(t)\,dt =\frac J{16} \sum_{r=1}^\infty\frac1{2^r(r+1)^2} <\frac18<1,

and even

\int_0^\infty t\phi(t)\,dt<\infty.

Take the moving-average kernel to be zero and any positive immigration rate. The continuous-time stationary point process exists by strict subcriticality.

Nevertheless, for every positive integer $m$ , take $\Delta=1/m$ . The sampling lattice contains every integer, and $\phi(n)=a_n$ . Therefore

\sum_{j\ge1}\Delta\phi(j\Delta) \ge \frac1m\sum_{n=2}^\infty a_n = \frac1m\sum_{r=1}^\infty\frac1{r+1} =\infty. \tag{2}

Hence there is no stationary sampled process for any of the arbitrarily fine meshes $\Delta=1/m$ , disproving the asserted existence of a universal sufficiently-small-mesh interval.

In fact nonexistence does not require assuming a finite stationary mean. If $\epsilon_i$ are the independent Poisson immigration counts, then $X_i\ge\epsilon_i$ . Since

\sum_na_n=\infty, \qquad \sum_na_n^2<\infty,

Kolmogorov's convergence criterion gives

\sum_{n\ge2}a_n\epsilon_{-mn}=\infty \quad\text{almost surely}.

The offspring contributions at lags $mn$ consequently have infinite total conditional Poisson intensity, so the putative finite-valued recursion is impossible almost surely.

A direct repair is to assume the stronger tail condition

\sum_{k\ge0} \sup_{t\in[k,k+1]}\phi(t)<\infty.

It makes the lattice-sum tails uniformly small and restores

\Delta\sum_{j\ge1}\phi(j\Delta) \longrightarrow\int_0^\infty\phi(t)\,dt.

An unconditional repair requiring only integrability is to replace point samples by cell averages

\widehat\phi_j(\Delta) =\int_{(j-1)\Delta}^{j\Delta}\phi(t)\,dt.

Then $\sum_j\widehat\phi_j=\int\phi<1$ for every mesh. Rounding the immigrant times and offspring delays in the same continuous-time branching trees gives exactly these cell-average offspring intensities; truncating after finitely many generations and using the geometric tail of the subcritical family proves vague convergence to the original point process as $\Delta\downarrow0$ .

Original withdrawn conjecture: Wheatley, Schatz, and Sornette, https://arxiv.org/abs/1806.09948 . Final published revision removing it: https://doi.org/10.1016/j.ecosta.2021.11.002 , Appendix A4. The same false automatic lattice-summability step appears in Kirchner, Hawkes and INAR(∞) processes, Lemma 1, https://doi.org/10.1016/j.spa.2016.02.008 .