Abstract
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions. We show that the limit is deterministic. As a consequence, we show that many intrinsic functionals of the embedded manifolds also converge to deterministic limits. Particularly interesting examples of these functionals are given by the Lipschitz-Killing curvatures, for which we also prove unbiasedness, using the Gaussian kinematic formula.
Highlights
We study the a.s. convergence of a sequence of random embeddings of a fixed manifold into Euclidean spaces of increasing dimensions
We show that many intrinsic functionals of the embedded manifolds converge to deterministic limits
In the recent paper [1] we studied the limiting behaviour of the global reach of a sequence of random manifolds embedded in Euclidean spheres of increasing dimensions
Summary
In the recent paper [1] we studied the limiting behaviour of the global reach of a sequence of random manifolds embedded in Euclidean spheres of increasing dimensions. The basic question was whether or not such functionals would converge to the corresponding intrinsic functional evaluated on (M, g), for an appropriately chosen metric g on M. Choosing g correctly, this turns out to be true, and the underlying reason is the fact that the Riemannian manifolds (hk(M ), gEk ) themselves converge, in an appropriate sense, to (M, g). We make the notions of “correctly” and “in an appropriate sense” precise, by describing some basic results on Gaussian processes and the convergence of manifolds. There we state the main result of the paper, Theorem 3.1, about the convergence of the random Riemannian manifolds (hk(M ), gEk ). We refer the interested reader to [1] for details
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