The DOI matches its Crossref record.
On Tail Probabilities for Martingales
Open accessAbstract
Watch a martingale with uniformly bounded increments until it first crosses the horizontal line of height $a$. The sum of the conditional variances of the increments given the past, up to the crossing, is an intrinsic measure of the crossing time. Simple and fairly sharp upper and lower bounds are given for the Laplace transform of this crossing time, which show that the distribution is virtually the same as that for the crossing time of Brownian motion, even in the tail. The argument can be adapted to extend inequalities of Bernstein and Kolmogorov to the dependent case, proving the law of the iterated logarithm for martingales. The argument can also be adapted to prove Levy's central limit theorem for martingales. The results can be extended to martingales whose increments satisfy a growth condition.
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
(2008) · ScholarlyCommons (University of Pennsylvania) · Cited by 581
Radner, R., & Rothschild, M. (1975) · Journal of Economic Theory · doi:10.1016/0022-0531(75)90006-x · Cited by 301
Minimax Regret Bounds for Reinforcement Learning
Azar, M. G., Osband, I., & Munos, R. (2017) · arXiv · doi:10.48550/arxiv.1703.05449 · Cited by 124
Metadata
- Authors
- Freedman, D. A.
- Venue
- The Annals of Probability
- Publisher
- Institute of Mathematical Statistics
- Published
- 1975-02-01
- Volume
- 3
- Issue
- 1
- ISSN
- 0091-1798 (ISSN-L)
- OpenAlex
- W2083459869
- Cited by
- 618
- References
- 0
- FWCI
- 4.72
- 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: On Tail Probabilities for Martingales. https://app.cochat.ai/papers/10.1214/aop/1176996452
- The paper
- Freedman, D. A. (1975). On Tail Probabilities for Martingales. The Annals of Probability. https://doi.org/10.1214/aop/1176996452
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.
