Asymptotics of Kantorovich Distance for Empirical Measures of the Laguerre Model

We estimate the rate of convergence for the Kantorovich (or Wasserstein) distance between empirical measures of i.i.d. random variables associated with the Laguerre model of order $\alpha$ on $(0,\mathrm{\infty }{)}^N$ and their common law, which is not compactly supported and has no rotational symmetry. Compared with the Gaussian case, our result is sharp provided the parameter $\alpha$ and the dimension $N$ are chosen in a specified regime.