We give sharp estimates of the probability that a given vector is in the convex hull of i.i.d. random vectors in a very general setting. When this given vector is the expectation, the problem is closely related to the problem of constructing cubature formulas in numerical analysis, and we give an answer to the following question: how many points do we need to sample before the random algorithm finishes? In the second part of the talk, we will show that, in some specific cases, the estimate can be improved by using hypercontractivity in the Gaussian Wiener chaos. This gives an insight into the random construction of cubature on Wiener space. This talk is mainly based on the paper https://arxiv.org/abs/2101.04250 (joint work with Terry Lyons and Harald Oberhauser).
Satoshi is a DPhil student under the supervision of Professor Terry Lyons at the University of Oxford. He completed his Master's in Information Science and Technology at the University of Tokyo.
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