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Estimating an even spherical measure from its sine transform

Lars Michael Hoffmann (2009)

Applications of Mathematics

To reconstruct an even Borel measure on the unit sphere from finitely many values of its sine transform a least square estimator is proposed. Applying results by Gardner, Kiderlen and Milanfar we estimate its rate of convergence and prove strong consistency. We close this paper by giving an estimator for the directional distribution of certain three-dimensional stationary Poisson processes of convex cylinders which have applications in material science.

Extremal points of high-dimensional random walks and mixing times of a brownian motion on the sphere

Ronen Eldan (2014)

Annales de l'I.H.P. Probabilités et statistiques

We derive asymptotics for the probability that the origin is an extremal point of a random walk in n . We show that in order for the probability to be roughly 1 / 2 , the number of steps of the random walk should be between e n / ( C log n ) and e C n log n for some constant C g t ; 0 . As a result, we attain a bound for the π 2 -covering time of a spherical Brownian motion.

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