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The conjecture on a stationary process without a good ARMA approximation

Let (ϵt)(\epsilon_t) be uncorrelated white noise random variables, and define the stationary process

Xt=ϵt−ϵt−12+ϵt−23+⋯=∑n=0∞(−1)nn+1ϵt−n.X_t=\epsilon_t-\frac{\epsilon_{t-1}}{2}+\frac{\epsilon_{t-2}}{3}+\cdots=\sum_{n=0}^{\infty}\frac{(-1)^n}{n+1}\epsilon_{t-n}.

For integers M,N≥0M,N\geq 0, consider an ARMA model

Yt=∑n=1NqnYt−n+∑n=0Mpnϵt−n,Y_t=\sum_{n=1}^{N}q_nY_{t-n}+\sum_{n=0}^{M}p_n\epsilon_{t-n},

with MM moving-average terms and NN autoregressive terms, and the truncation

Yt‾=∑n=0M+N(−1)nn+1ϵt−n.\overline{Y_t}=\sum_{n=0}^{M+N}\frac{(-1)^n}{n+1}\epsilon_{t-n}.

Conjecture on the absence of a good ARMA approximation. The process XtX_t does not have a good ARMA model of the displayed form that is better than the simple M+NM+N-term truncation (or moving-average approximation) Yt‾\overline{Y_t}.

Sources & referencesView supporting material

Primary source

Anand Ganesh, Babhrubahan Bose and Anand Rajagopalan, “On an L^2 norm for stationary ARMA processes”, arXiv:2408.10610 (2026).

arma-processesstationary-processestime-series

  • The conjecture is underspecified because “better” is undefined. If “better” refers to the L2L^2 approximation criterion introduced in the source paper, then the conjecture is refuted by the stable AR(1)(1) model

    Yt=ϵt−12Yt−1.Y_t=\epsilon_t-\tfrac12Y_{t-1}.
    — Nourdine Dehri

Progress summary

Refreshed
Open

A 2024 paper records the conjecture but leaves it open, and the scan found no later proof or counterexample.

The conjecture concerns whether the logarithmic-coefficient process can be approximated by an ARMA model more accurately than its corresponding finite truncation. It is explicitly presented as unresolved in the 2024 paper that formulates the example.

Known results

  • A rational/operator approximation theorem applies when lim sup⁡n→∞∣an∣1/n<1\limsup_{n\to\infty}|a_n|^{1/n}<1, but the example has a(z)=log⁡(1+z)a(z)=\log(1+z) and does not satisfy this condition.

August 2024 unresolved formulation

The paper states that the conjectured absence of a good ARMA approximation “requires further justification” and notes that its theorem cannot decide this example. The retrieved sources report no proof, counterexample, verification, or claimed AI-generated solution.

Current status (as of August 2026): The conjecture remains open; no proof, counterexample, or verification was found in the retrieved public sources.

Sources

Solutions 0

No solutions have been posted yet.