Tail-Dependence Estimators Under Long-Range Dependence

Definitions and mathematical statement

Let $X=(X_t)$ denote the stationary max-stable time series specified in the supplied record. Stationarity means that finite-dimensional distributions are invariant under admissible common time shifts; max-stability means that its finite-dimensional distributions are max-stable. The record does not specify the time index set, marginal distribution, or the additional hypotheses used in the source report.

For a deterministic threshold sequence $(u_n)_{n\geq 1}$, put \[ p_n:=\mathbb P(X_0>u_n). \] Assume \[ u_n\longrightarrow\infty, \qquad n p_n\longrightarrow\infty \qquad(n\longrightarrow\infty). \] The process is further required to be long-range dependent in the sense of the source report's Definition 1. That definition is absent from the supplied record, so this requirement cannot be replaced here by an explicit mathematical condition.

Write $\widehat\rho_{B,n}(T)$ for the empirical tail-dependence estimator named in the supplied statement. The record supplies neither its formula nor the meanings and admissible domains of $B$ and $T$. It also does not specify its population target, centering, or normalization.

Requested problem, subject to these unresolved definitions. Determine the asymptotic distribution of $\widehat\rho_{B,n}(T)$ in the displayed threshold regime for the source report's class of long-range-dependent stationary max-stable time series. This includes identifying any centering and normalization required to obtain a meaningful limiting law. Schematically, if a centered and scaled limit is intended, identify deterministic sequences $b_n\in{\mathbb{R}}$, $a_n>0$, and a probability law $L$ such that \[ a_n\left(\widehat\rho_{B,n}(T)-b_n\right) \xrightarrow{\mathrm d} L. \] This display describes a possible form of the requested limit theorem; the supplied record does not assert its existence, prescribe these sequences, or establish a common law for the whole class.

The original excerpt contrasts short-range asymptotic normality with an unresolved long-range limiting distribution. It also contains a corrupted variance expression involving $n\mathbb P(X_0>u_n)$ and a statistic rendered as $\widehat P_{n,u_n}((0,\infty)^{|T|})$. Its brackets, scaling, and limiting operation cannot be recovered reliably from the supplied text. Consequently, it does not furnish a precise additional hypothesis or variance conclusion for this draft.

Short English statement

Determine the limiting distribution, with any necessary centering and scaling, of the stated empirical tail-dependence estimator for a stationary max-stable time series with long-range dependence, when thresholds increase and the expected number of exceedances tends to infinity. A fully specified statement requires the source's estimator formula, parameter domains, and definition of long-range dependence.

Sources

• Supplied local problem registry record: permanent UnsolvedMath ID $30005968$, title Tail-Dependence Estimators Under Long-Range Dependence, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2024.
• Supplied source citation: Mathematics, Statistics, and Geometry of Extreme Events in High Dimensions (2025), Oberwolfach Reports, pp. 2179--2250, DOI 10.4171/owr/2024/38. Source URL: https://doi.org/10.4171/owr/2024/38. The background describes this report as documenting a mathematical workshop and its open-problem discussions.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2024/38.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially_solved, reports variance bounds and asymptotic normality in the short-range case, and describes the long-range limiting distribution as explicitly open. This is supplied dated curation derived from the OWR report, not an independently verified literature assessment; time-sensitive status and cited claims require independent checking.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not verification performed for this draft.