Marginal Limits for Markov-Source Selection Processes

Definitions and mathematical statement

The supplied record concerns the bucket-operation process $Y_n$ for rank selection from strings generated by a Markov source, where $n\to\infty$ is the asymptotic parameter. It does not define the bucket-operation cost, the sampling model, the rank parametrization, or the centering.

To express the intended question schematically, let $T$ denote the unspecified index set of the process, write $Y_n(t)$ for its value at $t\in T$, and write $c_n(t)$ for the intended, but unspecified, centering. Set \[ Z_n(t)=\frac{Y_n(t)-c_n(t)}{\sqrt n}. \] Here $T$ and $c_n$ are placeholders for definitions that must be recovered from the cited sources; they are not choices supplied by this draft.

The intended marginal-convergence question is whether, for each fixed admissible index $t$, there exists a Borel probability measure $\mu_t$ on ${\mathbb{R}}$ such that \[ \mathcal L\left(Z_n(t)\right)\Longrightarrow\mu_t \qquad\text{as }n\to\infty, \] where $\mathcal L(X)$ denotes the distribution of a real-valued random variable $X$, and $\Longrightarrow$ denotes weak convergence of probability measures. This asks about one-dimensional distributions at fixed indices; it does not specify joint or process convergence.

This is an incomplete formalization: the supplied record does not determine the admissible Markov sources, their alphabet, transition probabilities or initial distribution, the dependence assumptions for the input strings, the treatment of ties, or how an index determines an integer rank as $n$ varies. It also does not determine whether boundary ranks are included or whether exceptional indices require separate treatment. Thus the intended source quantifier and index domain cannot be stated precisely from the supplied evidence.

The supplied dated assessment labels the problem open and reports process non-tightness for general Markov sources while leaving marginal convergence open. These are supplied literature claims, not independently verified conclusions of this draft.

Short English statement

For the stated Markov-source rank-selection model, does the bucket-operation cost at each fixed admissible rank parameter converge in distribution after the intended centering and division by $\sqrt n$? The record leaves the model, rank domain and centering unspecified, so the displayed formulation is schematic and requires mathematical review.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30002720$, source problem code catalogue entry, title Marginal Limits for Markov-Source Selection Processes. The supplied record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2014, and names no proposer.
• Supplied report citation: Probability, Trees and Algorithms (2015), Oberwolfach Reports, DOI 10.4171/owr/2014/50, pp. 2825--2871. https://doi.org/10.4171/owr/2014/50. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Radix selection on Markov sources. https://doi.org/10.1016/j.spa.2018.03.009.
• Supplied literature source: Process convergence preprint. https://arxiv.org/abs/1605.02352.
• Literature assessment dated exactly 2026-08-21: the record labels the problem open and states that the primary process-convergence paper proves non-tightness for general Markov sources but explicitly leaves convergence of the marginals open. This is supplied dated curation derived from the supplied OWR curation, not an independently verified literature assessment. The time-sensitive status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified and reports a manual semantic, typographic and LaTeX review that regarded the cleanup as meaning-preserving and self-contained. That supplied review label does not resolve the missing model definitions identified here. No cited external source was accessed for this draft.