Open Problems on Smoothed Laplace Periodograms

Definitions and mathematical statement

A time series is a family of random observations indexed by time. The supplied record does not specify its state space, probability law, dependence assumptions, or sampling scheme. It also does not provide mathematical definitions of the Laplace periodogram, its rank-based variants, or the smoothing operation. These estimator names therefore retain their source-specific meanings, pending consultation of the cited report.

The supplied statement is the following research program:

1. Determine asymptotic distributions of smoothed Laplace periodograms and their rank-based variants as the sample size tends to infinity.
2. Develop and analyze applications of these methods to signal detection.
3. Extend these methods to locally stationary time series.

A precise distributional assertion requires definitions of the estimators, their frequency and quantile parameters, smoothing kernels and bandwidth sequences, assumptions on the underlying process, centering and normalization, and the intended mode of convergence. The signal-detection direction additionally requires a signal model, null and alternative hypotheses, and a performance criterion. The extension direction requires a definition of local stationarity and specifications of time localization and the target spectral quantities. None of these choices is supplied. Consequently, the record does not determine a single quantified conjecture or a uniquely specified asymptotic limit theorem; the three directions above must remain qualified research objectives.

Short English statement

Study the limiting distributions of smoothed Laplace periodograms and their rank-based variants, develop signal-detection applications, and extend the methods to locally stationary time series. The supplied record leaves the estimators and statistical assumptions unspecified. Its dated curation labels the program partially solved, without identifying a precise remaining conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002243$, title: Open Problems on Smoothed Laplace Periodograms; collection: Oberwolfach Reports Open Problems; exact source problem code: catalogue entry. No individual proposer is supplied. The proposed year is recorded as 2012.
• Supplied report citation: Mini-Workshop: Frontiers in Quantile Regression (2013), Oberwolfach Reports, DOI 10.4171/owr/2012/56, pp. 3339--3373. https://doi.org/10.4171/owr/2012/56. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. Its statement-verification field asserts that the official report explicitly labels these three directions as interesting open questions currently under investigation; its statement-status field labels the statement corrected and verified. These are supplied source-verification claims, not independent verification here. The citation's 2013 year differs from the recorded proposed year and the year appearing in the DOI; this discrepancy remains unresolved.
• Supplied literature source, labeled "quantile spectral processes": https://arxiv.org/abs/1401.8104.
• Supplied literature source, labeled "locally stationary quantile spectra": https://academic.oup.com/jrsssb/article/79/5/1619/7040912.
• Supplied literature source, labeled "periodic signal detection application": https://pmc.ncbi.nlm.nih.gov/articles/PMC2650682/.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the status partially solved and reports subsequent developments in smoothed Laplace periodogram asymptotics, signal-detection applications, and locally stationary extensions. It characterizes the record as a broad research program rather than a single completely closed theory. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification.