A Gaussian Correlation Inequality for Symmetric Convex Sets
Abstract
If $n(x)$ is the standard normal density on $R^2$ and if $A = -A$ and $B = -B$ are convex subsets of $R^2$ then $$\int_{A\cap B}\mathbf{n}(x) d^2x \geqq (\int_A \mathbf{n}(x) d^2x)(\int_B \mathbf{n}(x) d^2x).$$
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
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.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
Pitt's Gaussian correlation inequality gives the qualitative bound γW ≥ 0 but not a quantitative comparison to γ.
“Pitt's Gaussian correlation inequality (Pitt, 1977;Esary et al., 1967;Joag-Dev and Proschan, 1983) gives the qualitative bound γW ≥ 0 but not a quantitative comparison to γ, so direct estimation is the right move.”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Pitt, 1977
Theorem 3 and its remarks (p. 3) establish a qualitative Gaussian correlation bound: the joint probability is nondecreasing from the product of the marginal probabilities. But the paper does not define γW or γ, so our copy cannot establish that its bound is specifically γW ≥ 0 or assess a comparison with γ. (LLM, conf=0.82)
Experimental — accuracy not yet measuredFor jointly Gaussian competent noise with positive score correlation, the cluster indicators are positively associated by Pitt's Gaussian correlation inequality, so γW ≥ 0.
“For jointly Gaussian competent noise with positive score correlation, the cluster indicators are positively associated by Pitt's Gaussian correlation inequality (Pitt, 1977;Esary et al., 1967;Joag-Dev and Proschan, 1983), so γW ≥ 0”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Pitt, 1977
Theorem 3 and its remarks (p. 3) imply nonnegative association for indicators of balanced convex sets under the stated Gaussian covariance conditions, including a cross-covariance rank of at most two. The paper does not establish that the citing document’s cluster indicators meet those conditions, or address γW; positive score correlation alone does not supply the missing link in this source. (LLM, conf=0.86)
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Cites
No references could be resolved yet.
Research works printed but not linked: 4
- [1][1] ANDERsON, T. W. (1958). An Introduction to Multivariate Statistical Analysis. Wiley, New York. -
- [2][2] ANTELL, R. H. (1975). The normal probability content of convex symmetric sets. Master's thesis, Univ. of Virginia. -
- [4][4] DAS GUPTA, S., EATON, M. L., OLKIN, I., PERLMAN, M., SAVAGE, L. J. and SOBEL, M. (1972). Inequalities on the probability content of convex regions for elliptically contoured distributions. Proc. Sixth Berkeley Symp. Math. Statist. Prob. 2 241-264. Univ. of California Press. -
- [6][6] MILNOR, J. (1965). Topology for the Differentiable Viewpoint. Univ. Press of Virginia, Charlottesville. -
Cited by
Sorted by citations
RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Pitt, L. D.
- Venue
- The Annals of Probability
- Publisher
- Institute of Mathematical Statistics
- Published
- 1977-06-01
- Volume
- 5
- Issue
- 3
- Pages
- 470-474
- ISSN
- 0091-1798
- 2168-894X
- Cited by
- 42
- References
- 0
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
