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Department of Mathematical Sciences

Seminar Archives

On this page you can find information about seminars in this and previous academic years, where available on the database.

Statistics Seminars: Geometric properties of convex hulls of random walks

Presented by Vladislav Vysotsky, Arizona State University, Imperial College London, and Steklov Mathematical Institute St. Petersburg

19 October 2015 14:00 in CM221

Our work was motivated by the following question: What is the
probability that the convex hull of an n-step random walk in R^d does
not contain the origin? In dimension one this is simply the probability
that the walk does not change its sign by the time n. The remarkable
formula by Sparre Andersen (1949) states that a random walk with
symmetric density of increments stays positive with probability
(2n-1)!!/(2n)!! Our result is a two-dimensional distribution-free
counterpart of this fact. The developed approach is then used to study
other geometric characteristics of convex hulls of random walks. We
obtain results on the expected number of faces, volume, total surface
area of faces, and other intrinsic volumes of the convex hull. If time
allows, I will briefly mention a different technique that solves the
original problem in general dimension.

This is a joint work with Dmitry Zaporozhets (St. Petersburg)

Contact andrew.wade@durham.ac.uk for more information