The DOI matches its Crossref record.
Strong Consistency of $K$-Means Clustering
Open accessAbstract
A random sample is divided into the $k$ clusters that minimise the within cluster sum of squares. Conditions are found that ensure the almost sure convergence, as the sample size increases, of the set of means of the $k$ clusters. The result is proved for a more general clustering criterion.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cited by
Consistency of spectral clustering
von Luxburg, U., Belkin, M., & Bousquet, O. (2008) · The Annals of Statistics · doi:10.1214/009053607000000640 · Cited by 540
Grouped Patterns of Heterogeneity in Panel Data
Bonhomme, S., & Manresa, E. (2015) · Econometrica · doi:10.3982/ecta11319 · Cited by 451
Metadata
- Authors
- Pollard, D.
- Venue
- The Annals of Statistics
- Publisher
- Institute of Mathematical Statistics
- Published
- 1981-01-01
- Volume
- 9
- Issue
- 1
- ISSN
- 0090-5364 (ISSN-L)
- OpenAlex
- W2086943813
- Cited by
- 454
- References
- 0
- FWCI
- 5.79
- Type
- journal article
- Language
- en
- Open access
- bronze
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Strong Consistency of $K$-Means Clustering. https://app.cochat.ai/papers/10.1214/aos/1176345339
- The paper
- Pollard, D. (1981). Strong Consistency of $K$-Means Clustering. The Annals of Statistics. https://doi.org/10.1214/aos/1176345339
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
