Approximation conjecture for the ARMA point process by INARMA processes

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.

Progress summary

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

arxiv.org

Sources & referencesView supporting material

Primary source

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

1 week ago

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 .

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Shivam Patel · 2 days ago