We prove that the Dirichlet form associated with the Wasserstein diffusion on the set of all probability measures on the unit interval, introduced in von Renesse and Sturm (Entropic measure and Wasserstein diffusion. Ann Probab, 2008) satisfies a logarithmic Sobolev inequality. This implies hypercontractivity of the associated transition semigroup. We also study functional inequalities for related diffusion processes.
Mathematics Subject Classification (2000)Primary: 58J65 · 47D07; Secondary: 28A33 · 35P15 · 60J35
We consider the problem of estimating the location of an asymmetric cusp θ 0 in a regression model. That means, we focus on regression functions, which are continuous at θ 0 , but the degree of smoothness from the left p 0 could be different to the degree of smoothness from the right q 0 . The degrees of smoothness have to be estimated as well. We investigate the consistency with increasing sample size n of the leastsquares estimates. It turns out that the rates of convergence ofθ n depend on the minimum b of p 0 and q 0 and that our estimator converges to a maximizer of a Gaussian process. In the regular case, that is, for b greater than 1 2 , we have a rate of √ n and the asymptotic normality property. In the non-regular case, we have a representation of the limit distribution ofθ n as maximizer of a fractional Brownian motion with drift.
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