# The topology of probability distributions on manifolds

@article{Bobrowski2013TheTO,
title={The topology of probability distributions on manifolds},
author={Omer Bobrowski and Sayan Mukherjee},
journal={Probability Theory and Related Fields},
year={2013},
volume={161},
pages={651-686}
}
• Published 3 July 2013
• Mathematics
• Probability Theory and Related Fields
Let $$\mathcal {P}$$P be a set of $$n$$n random points in $$\mathbb {R}^d$$Rd, generated from a probability measure on a $$m$$m-dimensional manifold $$\mathcal {M}\subset \mathbb {R}^d$$M⊂Rd. In this paper we study the homology of $$\mathcal {U}(\mathcal {P},r)$$U(P,r)—the union of $$d$$d-dimensional balls of radius $$r$$r around $$\mathcal {P}$$P, as $$n\rightarrow \infty$$n→∞, and $$r\rightarrow 0$$r→0. In addition we study the critical points of $$d_{\mathcal {P}}$$dP—the distance function…
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