Convergence Radii of Autoregressive and Moving Average Persistence Series

Definitions and mathematical statement

For $T\in\{\mathrm{AR}(1),\mathrm{MA}(1)\}$, write $\theta_T(\rho)$ for the persistence-exponent function intended in the supplied source for the corresponding Gaussian process. The source does not specify the process equations, the admissible real parameter range, the initial distribution, the persistence event, or the convention for the persistence exponent. Consequently, these symbols do not yet identify unique functions.

For example, once a process law and a zero-level persistence event have been specified, one may define \[ p_{T,n}(\rho)=\mathbb P_\rho\left(X_1^{(T)}>0,\ldots,X_n^{(T)}>0\right). \] Two quantities that must be distinguished are the logarithmic decay rate \[ -\lim_{n\to\infty}\frac{1}{n}\log p_{T,n}(\rho) \] and the exponential survival rate \[ \lim_{n\to\infty}p_{T,n}(\rho)^{1/n}, \] whenever the respective limits exist. This example explains a convention requiring clarification; it does not establish the intended persistence event or exponent convention.

If the intended power series are expansions about $\rho=0$, write them as \[ \theta_T(\rho)=\sum_{k=0}^{\infty}a_{T,k}\rho^k. \] Their radii of convergence are \[ R_T=\sup\left\{r\geq0:\ \sum_{k=0}^{\infty}|a_{T,k}|s^k<\infty \text{ for every }0\leq s<r\right\} =\left(\limsup_{k\to\infty}|a_{T,k}|^{1/k}\right)^{-1}, \] with $1/0=\infty$ and $1/\infty=0$. This definition concerns convergence for complex $\rho$, even if the underlying stochastic model is initially defined only for real parameters.

The requested problem is to determine $R_{\mathrm{AR}(1)}$ and to decide whether \[ R_{\mathrm{MA}(1)}=1. \] The expansion centre, model parameterizations, exponent convention, and existence of the intended expansions require confirmation from the cited sources before this conditional formulation becomes a uniquely specified mathematical problem. The supplied record labels the problem open; its dated assessment reports only lower bounds for the autoregressive radius and describes the moving-average radius-one assertion as conjectural. Those are supplied curation claims, not independently verified mathematical conclusions.

Short English statement

Determine the convergence radius of the Gaussian AR(1) persistence-exponent series and decide whether the analogous MA(1) series has radius one. The supplied record leaves the precise process parameterizations, persistence-exponent convention, and expansion centre unspecified, so these must be confirmed before the target series are uniquely determined.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004539$, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is identified.
• Supplied report citation: Stochastic Processes under Constraints (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/32, pp. 1601--1656. https://doi.org/10.4171/owr/2020/32. The registry separately records proposed year $2020$.
• Supplied literature source, labelled "AR(1) persistence series": https://arxiv.org/abs/1810.09861.
• Supplied literature source, labelled "MA(1) persistence series": https://arxiv.org/abs/2407.06870.
• Supplied literature source, labelled "OWR report": https://ems.press/content/serial-article-files/46870.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, reports only lower bounds on the AR(1) convergence radius in the cited work, and describes the MA(1) radius-one assertion as conjectural, with no complete proof of the requested radius available. This is supplied dated curation derived from the OWR record, not independently verified; time-sensitive status and cited claims require independent checking.