Gaussian Limts for Random Geometric Measures

Gaussian limits for random geometric measures

Given n independent random marked d-vectors Xi with a common density, define the measure νn = ∑ i ξi, where ξi is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near Xi. Technically, this means here that ξi stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions f on R, we give a central l...

Matrix measures, random moments and Gaussian ensembles

We consider the moment space Mn corresponding to p × p real or complex matrix measures defined on the interval [0, 1]. The asymptotic properties of the first k components of a uniformly distributed vector (S1,n, . . . , Sn,n) ∗ ∼ U(Mn) are studied if n → ∞. In particular, it is shown that an appropriately centered and standardized version of the vector (S1,n, . . . , Sk,n) ∗ converges weakly to...

Convergence of random measures in geometric probability

Given n independent random marked d-vectors Xi with a common density, define the measure νn = ∑ i ξi, where ξi is a measure (not necessarily a point measure) determined by the (suitably rescaled) set of points near Xi. Technically, this means here that ξi stabilizes with a suitable power-law decay of the tail of the radius of stabilization. For bounded test functions f on Rd, we give a law of l...

Autoregressive Gaussian Random Vectors of First Order

Gaussian Marginals of Probability Measures with Geometric Symmetries

Let K be a convex body in the Euclidean space Rn, n ≥ 2, equipped with its standard inner product 〈·, ·〉 and Euclidean norm | · |. Consider K as a probability space equipped with its uniform (normalized Lebesgue) measure μ. We are interested in k-dimensional marginals of μ, that is, the push-forward μ◦P−1 E of μ by the orthogonal projection PE onto a k-dimensional subspace E ⊂ Rn. The question ...