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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 $d\mu (x) = C_{d} (1-x^{2})^{\frac d2 -1} dx$ with dimension d ≥ 1 on [− 1,1], and let $\mu _{n} = \frac {1}{n} {\sum }_{i = 1}^{n} \delta _{X_{i}}$ be the associated empirical measure. We show that $$ \lim\limits_{n \to \infty} n{\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \sum\limits_{k = 1}^{\infty} \frac{1}{k(k+d-1)}, $$ where W2 is the quadratic Kantorovich distance with respect to the intrinsic cost $\rho (x, y) = |\arccos (x) - \arccos (y)|$ , (x,y) ∈ [− 1,1]2, associated to the model. When μ is the product of two Jacobi measures with dimensions d and $d^{\prime }$ respectively, then $$ {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] \approx \frac{\log n}{n} . $$ In the particular case $d = d^{\prime } = 1$ (corresponding to the product of arcsine laws), $$ \lim_{n \to \infty} \frac{n}{\log n} {\mathbb{E}} \left[ {W_{2}^{2}}(\mu^{n}, \mu ) \right] = \frac{\pi}{4}. $$ 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.

Consider the direct product manifold $$M_1 \times \cdots \times M_n$$ , where $$M_i$$ ( $$1 \le i \le n$$ ) are connected complete non-compact Riemannian manifolds satisfying the volume doubling property and Gaussian or sub-Gaussian estimates for the heat kernel. We obtain weak type (1, 1) (so $$L^p$$ -boundedness with $$1< p < 2$$ ) for the Riesz transform. As a consequence, we find that neither heat kernel Gaussian estimates nor sub-Gaussian estimates are necessary for weak (1, 1) property of Riesz transform.

This article is a continuation of the papers [8,9] in which the optimal matching problem, and the related rates of convergence of empirical measures for Gaussian samples are addressed. A further step in both the dimensional and Kantorovich parameters is achieved here, proving that, given $X_1, \ldots, X_n$ independent random variables with common distribution the standard Gaussian measure $\mu$ on $\mathbb{R}^d$, $d \geq 3$, and $\mu_n \, = \, \frac 1n \sum_{i=1}^n \delta_{X_i}$ the associated empirical measure, $$ \mathbb{E} \big( \mathrm {W}_p^p (\mu_n , \mu )\big ) \, \approx \, \frac {1}{n^{p/d}} $$ for any $1\leq p < d$, where $\mathrm {W}_p$ is the $p$-th Kantorovich metric. The proof relies on the pde and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan in a compact setting.