2019/03/28 by Jie-Xiang Zhu, Jiexiang Zhu, Zhu, Jiexiang · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math.PR
paper · pdf · doi:10.48550/arxiv.1903.11739
openalex publication_date 2019/03/28 · arxiv created 2019/11/24 · arxiv updated 2019/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish some exact asymptotic results for a matching problem with respect to a family of beta distributions. Let X1, …, Xn be independent random variables with common distribution the symmetric Jacobi measure dμ(x) = Cd (1-x2)\frac d2 -1 dx with dimension d ≥ 1 on [-1, 1], and let μn = (1)/(n) ∑i = 1n δXi be the associated empirical measure. We show that limn → ∞ n\E [ W22( μn, μ) ] = ∑k = 1∞ (1)/(k(k+d-1)), where W2 is the quadratic Kantorovich distance with respect to the intrinsic cost ρ(x, y) = |\arccos(x) - \arccos (y)|, (x, y) ∈ [-1, 1]2, associated to the model. When μ is the product measure of two Jacobi measures with dimensions d and d' respectively, then \E [ W22( μn, μ) ] ≈ (log n)/(n). In the particular case d = d' = 1 (corresponding to the product of arcsine laws), limn → ∞ (n)/(log n) \E [ W22( μn, μ) ] = \fracπ4. Similar results do hold for non-symmetric Jacobi distributions. The proofs are based on the recent PDE and mass transportation approach developed by L.~Ambrosio, F.~Stra and D.~Trevisan.